Lagrange’s four squares theorem with one prime and three almost–prime

variables

D. R. Heath–Brown  D. I. Tolev

1 Introduction and statement of the results.

In this paper we study the equation of Lagrange

x_{1}^{2} + x_{2}^{2} + x_{3}^{2} + x_{4}^{2} = N (1)

with multiplicative restrictions imposed on the variables.

Various proofs for the solvability of (1) in integers x_{1}, x_{2}, x_{3}, x_{4} are known. We should mention the explicit formula for the number of integer solutions of (1), discovered by Jacobi. We refer the reader to the books of Hardy and Wright [6] and Hua [10], for example.

Kloosterman [15] considered the problem of the representation of large integers N by the integral positive definite quadratic form

a_{1}x_{1}^{2} + a_{2}x_{2}^{2} + a_{3}x_{3}^{2} + a_{4}x_{4}^{2} .

His method yields an asymptotic formula for the number of representations. The classical circle method of Hardy, Littlewood and Ramanujan provides an asymptotic formula for quadratic forms with five or more variables only.

It is expected that every sufficiently large integer N, satisfying the congruence condition N ≡ 4 (mod 24), can be represented in the form (1) with prime variables x_{i}. This hypothesis has not been proved so far. Using Vinogradov’s method for the solution of the ternary Goldbach problem, however, Hua [9] proved that all large integers, satisfying a natural congruence condition, are sums of five squares of primes.

Greaves [5], Plaksin [16] and Shields [17] proved the solvability of (1) with two prime and two integer variables. Actually, in [16] and [17] an asymptotic formula for the number of solutions was found.

Brüdern and Fouvry [1] considered (1) with sufficiently large N satisfying N ≡ 4 (mod 24) and found a lower bound for the number of solutions in interges x_{i} with no more than 34 prime factors each.

The main purpose of the present paper is to prove the following:

Theorem 1. Every sufficiently large integer N, satisfying N ≡ 4 (mod 24), can be represented in the form

p^{2} + x_{1}^{2} + x_{2}^{2} + x_{3}^{2} = N, (2)

where p is a prime and x_{i} are integers without prime factors less than N^{0.004915}. The number of such representations exceeds cN(log N)^{−4} for some positive constant c. In particular, every such x_{i} has at most 101 prime factors.

We are also in a position to improve slightly the result of Brüdern and Fouvry from [1]. We have

Theorem 2. Every sufficiently large integer N, satisfying N ≡ 4 (mod 24), can be represented in the form (1), where x_{i} are integers without prime factors less than N^{0.01995}. The number of such representations exceeds cN(log N)^{−4} for some positive constant c. In particular, every such x_{i} has at most 25 prime factors.

In spite of the results of Greaves, Plaksin and Shields already referred to, there seems little hope at present of establishing a version of Lagrange’s Theorem involving two primes and two almost-primes. Although we can show that N − p^{2} − q^{2} is a sum of two squares, we are unable to control sufficiently the divisibility properties of the two squares that arise. To prove Theorem 1 we have to show that N − p^{2} is a sum of three squares, and to control the distribution in residue classes of these three squares. A standard application of the circle method, with the Kloosterman refinement, is not quite sufficient for this purpose. However by using the first author’s “square sieve” [7], we are able to take advantage of the fact that we are considering numbers N − p^{2} in which p^{2} is a square, and this leads to a suitable saving. The usual machinery of the Kloosterman refinement is needed, but it appears that, at one point, when we consider the second estimate for E(G_{1}, G_{2}) in §3.4.6, this is insufficient. This rather technical problem is overcome by using a double Kloosterman refinement. Thus, if the Farey dissection in our application of the circle method involves arcs around the points a/q, we average non-trivially not only over the numerators a, but also over the denominators q. It would seem that this is the first occasion on which such a technique has been successfully employed to give an unconditional result.

Acknowledgements. The main part of this research was done during the visit of the second author to the Mathematical Institute of the University of Oxford. The second author thanks the Royal Society for financial support, the staff of the Institute for the excellent working conditions and also Plovdiv University Scientific Fund (grant PU2–MM) for covering some other expenses.

2 Notations and some Lemmas.

Denote P = N^{1/2}. Let ε ∈ (0, 10^{−6}) be an arbitrarily small fixed number. Let A > 10^{6} be an arbitrarily large number, which may not be the same in different occurrences. If it is not stated explicitly, the constants in O–terms and ≪–symbols are absolute or depend on ε and A. The letter p is reserved for prime numbers. In all sections, with the exception of Section 5, we denote by bold style letters three–dimensional vectors. In Section 5 we use bold style letters to denote four–dimensional vectors. We denote by (m_{1}, ⋯, m_{k}) and [m_{1}, ⋯, m_{k}] the greatest common divisor and, respectively, the least common multiple of the integers of m_{1}, ⋯, m_{k}. However, if u, v are real numbers then (u, v) means the interval with endpoints u and v. The meaning is always clear from the context.

As usual, μ(n), φ(n), τ(n), ν(n) denote the Möbius function, Euler’s function, the number of divisors of n and the number of distinct prime factors of n, respectively. Instead of m ≡ n (mod k) we write for simplicity m ≡ n (k) . If p^{l} | m, but p^{l+1} ∤ m then we write p^{l} || m. We denote e(α) = e^{2πiα} and e_{q}(α) = e(α/q). By [t] we denote the integer part of the real number t and let ||t|| be the distance from t to the nearest integer. We use ∑_{x(q)} and ∑_{x(q)^{∗}} to denote sums with x running over a complete system, respectively reduced system of residues modulo q. If q is an odd integer then by (\frac{l}{q} ) we denote the

Jacobi symbol. If (a, q) = 1 then \overline{(a)}_{q} means the inverse of a modulo q. If the value of the modulus is clear from the context then we simply write \overline{a}. For example, e_{q}(\overline{a}) always means e_{q}(\overline{(a)}_{q}) . For any a we put e_{1}(\overline{a}) = 1 . If U and V are positive then U ≍ V means that U ≪ V ≪ U. We use □ to mark an end of a proof or its absence.

We denote by π(x, q, m) the number of primes p ≤ x such that p ≡ m (q) and let

Δ(x, q, m) = π(x, q, m) − \frac{1}{φ(q)} ∫_{2}^{x} \frac{dt}{log t} . (3)

Our first lemma is a weak version of the Barban–Davenport–Halberstam theorem (see Chapter 29 of [3], for example).

Lemma 1. Suppose that Q ≪ x^{1−ε}. Then we have

∑_{q≤Q} ∑_{m(q)^{∗}} Δ(x, q, m)^{2} ≪ x^{2}(log x)^{−A} .

□

In our study we use the properties of the Kloosterman sum K(q, m, n), Ramanujan sum c_{q}(n) and the Gauss sums γ(q), S(q, a, m), S(q, a) and T (q, a).

These sums are defined by

(4)  K(q, m, n) = ∑_{x(q)^{∗}} e_{q}(mx + n\overline{x}) ,  c_{q}(m) = K(q, m, 0) ,

(5)  γ(q) = ∑_{l(q)} (\frac{l}{q} ) e_{q}(l)  ( for odd integers q only) ,
(6)  S(q, m, n) = ∑_{x(q)} e_{q}(mx^{2} + nx) ,  S(q, m) = S(q, m, 0) ,
(7)  T (q, m) = ∑_{x(q)^{∗}} e_{q}(mx^{2}) .

If d ∈ ℕ^{3} and n ∈ ℤ^{3} then we denote

S_{d}(q, m, n) = ∏_{i=1}^{3} S(q, m d_{i}^{2}, n_{i}) , S_{d}(q, m) = S_{d}(q, m, 0) . (8)

In Section 5, however, bold style vectors denote four–dimensional vectors and the definition of S_{d}(q, m, n) is different.

We also put

h_{d}(q) = q^{−3}φ(q)^{−1} ∑_{a(q)^{∗}} S_{d}(q, a) T (q, a) e_{q}(−aN) . (9)

In the next lemma we present some identities and inequalities for the sum S(q, m, n). The proof of (i) is elementary. The proofs of (ii) – (vi) are available in Section 6 of [4] and Chapter 7 of [10].

Lemma 2. The Gauss sum S(q, m, n) satisfies:

(i) If (q_{1}, q_{2}) = 1 then

S(q_{1}q_{2}, a_{1}q_{2} + a_{2}q_{1}, n) = S(q_{1}, a_{1}q_{2}^{2}, n) S(q_{2}, a_{2}q_{1}^{2}, n) .

(ii) Suppose that (q, m) = d. We have

S(q, m, n) = \begin{cases}dS(q/d, m/d, n/d) & if d | n , \\ 0 & if d ∤ n . \end{cases}

(iii) If (q, m) = 1 then |S(q, m, n)| ≤ 2q^{1/2}.

(iv) If (q, 2m) = 1 then

S(q, m, n) = e_{q}( − \overline{(4m)} n^{2}) (\frac{m}{q}) S(q, 1) .

(v) If (q, 2) = 1 then

S(q, 1) = \begin{cases}q^{1/2} & if q ≡ 1 (4) , \\ iq^{1/2} & if q ≡ −1 (4) . \end{cases}

(vi) If (a, 2) = 1 then

S(2^{l}, a) = \begin{cases}0 & if l = 1 , \\ 2^{l/2}(1 + i^{a}) & if l is even , \\ 2^{(l+1)/2}e(a/8) & if l > 1 and odd . \end{cases}

□

In the next lemma we study T (q, m). Again the proof of (i) is elementary. Identities (ii) and (iii) can be easily verified using definition (7) and Lemma 2.

Lemma 3. The Gauss sum T (q, m) satisfies:

(i) If (q_{1}, q_{2}) = 1 then

T (q_{1}q_{2}, a_{1}q_{2} + a_{2}q_{1}) = T (q_{1}, a_{1}) T (q_{2}, a_{2}) .

(ii) If p > 2 is a prime and (p, a) = 1 then

T (p^{l}, a) = \begin{cases}S(p, a) − 1 & if l = 1 , \\ 0 & if l > 1 . \end{cases}

(iii) If (a, 2) = 1 then

T (2^{l}, a) = \begin{cases}−1 & if l = 1 , \\ 2e(a/4) & if l = 2 , \\ 4e(a/8) & if l = 3 , \\ 0 & if l > 3 . \end{cases}

□

In the next lemma we present Weil’s estimate for the Kloosterman sum.

Lemma 4. We have

|K(q, m, n)| ≤ 2^{ν(q)}(2q)^{1/2} (m, n, q)^{1/2} .
□

A simple identity and an estimate for the Ramanujan sum are given below. A proof is available in Chapter 16 of [6].

Lemma 5. We have

c_{q}(n) = \frac{φ(q)}{φ (\frac{q}{(q,n)})} μ (\frac{q}{(q, n)}) .

In particular

|c_{q}(n)| ≤ (q, n) .

□

In the next lemma we give some properties of the Gauss sum γ(q). Proofs can be found in Chapter 7 of [10].

Lemma 6.

(i) For any odd integer q we have |γ(q)| ≤ q^{1/2}.

(ii) If p > 2 is a prime then γ(p) = S(p, 1).

□

In the next lemma we present the fundamental properties of the Jacobi symbol. A proof is available, for example, in Chapter 3 of [10].

Lemma 7. If q and q_{1} are odd integers and (q, q_{1}) = 1 then we have

(i)  (\frac{q}{q_{1}} )(\frac{q_{1}}{q} ) = (−1)^{\frac{q−1}{2} · \frac{q_{1}−1}{2}} ,
(ii)  (\frac{2}{q}) = (−1)^{\frac{q^{2}−1}{8}} ,
(iii)  (\frac{−1}{q}) = (−1)^{\frac{q−1}{2}} .
□

Now we shall study the function h_{d}(q), defined by (9). For any prime p > 2 we put

(10)  h_{0}(p) = \begin{cases}\frac{1}{p} & if p | N , \\ \frac{−1}{p(p−1)}(1 + (\frac{−N}{p} ) ) & if p ∤ N ; \end{cases}

(11)  h_{1}(p) = \begin{cases}\frac{−1}{p}(\frac{−1}{p}) & if p | N , \\ \frac{1}{p−1}((\frac{−N}{p} ) + \frac{1}{p}(\frac{−1}{p}) ) & if p ∤ N ; \end{cases}

(12)  h_{2}(p) = \begin{cases}(\frac{−1}{p}) & if p | N , \\ \frac{−1}{p−1}((\frac{−1}{p}) + (\frac{N}{p} ) ) & if p ∤ N ; \end{cases}

(13)  h_{3}(p) = \begin{cases}−1 & if p | N , \\ \frac{1}{p−1}( p(\frac{N}{p} ) + 1 ) & if p ∤ N . \end{cases}

The following lemma holds:

Lemma 8. Suppose that d_{1}, d_{2}, d_{3} are squarefree odd integers and N ≡ 4 (24). The function h_{d}(q) is multiplicative with respect to q. We have

(14) h_{d}(2^{l}) = \begin{cases}0 & if l = 1 , \\ −1/2 & if l = 2 , \\ 1/2 & if l = 3 , \\ 0 & if l > 3 . \end{cases}

If p > 2 is a prime then

(15) h_{d}(p^{l}) = \begin{cases}h_{j}(p) & if p^{j} || d_{1}d_{2}d_{3} and l = 1 , \\ 0 & if l > 1 . \end{cases}

The series Σ_{0}, defined below, is absolutely convergent and

Σ_{0} = Σ_{0}(d, N) = ∑_{q=1}^{∞} h_{d}(q) = ∏_{p>2}(1 + h_{d}(p)) . (16)

Proof: The multiplicativity of h_{d}(q) with respect to q follows easily from Lemmas 2 (i) and 3 (i). Formulas (14) and (15) are consequences of Lemma 2 (v), (vi) and Lemma 3 (ii), (iii). From (10) – (15) we find that the following estimate holds:

h_{d}(q) ≪ μ^{2}(\frac{q}{(q, 4)}) \frac{6^{ν(q)}}{q^{2}} (q, N) (d_{1}d_{2}d_{3})^{3} . (17)

Hence the series Σ_{0} is absolutely convergent and applying Euler’s identity we get (16).

□

Consider the function

ω_{0}(t) = \begin{cases}exp (\frac{1}{(20t−10)^{2}−1}) & if t ∈ (\frac{9}{20}, \frac{11}{20}) , \\ 0 & otherwise . \end{cases}

It is infinitely differentiable on the real line. Denote

ω(x) = ω_{0}(xP^{−1}) , ω(x) = ω(x_{1})ω(x_{2})ω(x_{3}) (18)

and

I(β, u) = ∫_{−∞}^{∞} ω_{0}(x) e(βx^{2} + ux)dx , I(β) = I(β, 0). (19)

If d ∈ ℕ^{3} and u ∈ ℝ^{3} then we define

I_{d}(β, u) = ∏_{i=1}^{3} I(β, u_{i}d_{i}^{−1}) . (20)

In Section 5, however, the definition of I_{d}(β, u) is different.

Lemma 9. The following estimates hold:

(i)  I(β, u) ≪ (1 + |β|^{k}) |u|^{−k}  for u ≠ 0 and for any k ∈ ℕ ,
(ii)  I(β) ≪ min (1, |β|^{−k})  for any k ∈ ℕ ,
(iii) I(β, u) ≪ min (1, |β|^{−1/2} ) .

The constants in ≪ – symbols in (i) and (ii) depend on k .

Proof: We prove (i) by multiple partial integration. To prove (ii) we change the variable x^{2} = y and then proceed as in the proof of (i) . Finally, the estimate (iii) is well–known.

□

Lemma 10.

(i) Suppose that u = max(|u_{1}|, …, |u_{6}|) > 0. Then we have

J = ∫_{−∞}^{∞} | I(β, u_{1}) … I(β, u_{6}) | dβ ≪ u^{−2+ε} .

(ii) Suppose that v = max(|v_{1}|, …, |v_{4}|) > 0. Then we have

∫_{−∞}^{∞} | I(β, v_{1}) … I(β, v_{4}) | dβ ≪ v^{−1+ε} .

Proof: We prove only (i). The proof of (ii) is similar. If u ≤ 1 the last inequality is a consequence of the trivial estimate J ≪ 1 . Suppose that u > 1 . We have J = J_{1} + J_{2} , where in J_{1} we integrate over |β| ≤ u^{1−ε/2} and in J_{2} over the other β . Lemma 9 (iii) implies that J_{2} ≪ ∫_{u^{1−ε/2}}^{∞} β^{−3} dβ ≪ u^{−2+ε} .

Now consider J_{1} . Suppose that u = |u_{1}| . We take the integer k = [6ε^{−1}] . If |β| ≤ |u_{1}|^{1−ε/2} then Lemma 9 (i) gives I(β, u_{1}) ≪ (1 + |β|^{k}) |u_{1}|^{−k} ≪ |u_{1}|^{(1−ε/2)k−k} . Therefore J_{1} ≪ |u_{1}|^{(1−ε/2)(k+1)−k} ≪ |u_{1}|^{−2} and the result follows.

□

Using Lemma 9 we see that the Fourier thansform of I^{3}(β)

H(t) = ∫_{−∞}^{∞} I^{3}(β) e(−tβ)dβ (21)

is uniformly and absolutely convergent. We have

Lemma 11. The function H(t) is non–negative and infinitely many times differentiable. It is supported and positive in (t_{1}, t_{2}) for some t_{1}, t_{2} such that 0 < t_{1} < t_{2} < 1. We have

I^{3}(β) = ∫_{−∞}^{∞} H(t) e(βt)dt (22) and

κ_{0} = ∫_{−∞}^{∞} |I(β)|^{6} dβ = ∫_{−∞}^{∞} H^{2}(t) dt . (23)

The integral

κ_{1} = ∫_{−∞}^{∞} I^{4}(β) e(−β) dβ (24)

is absolutely convergent and the constant κ_{1}, defined above, is real and positive. We also have

∫_{−∞}^{∞} H^{2}(t) e(βt)dt ≪ min(1, |β|^{−k}) (25)

for any integer k . The constant in ≪ – symbol depends on k.

Proof: The smoothness of H(t) as well as the identities (22) and (23) are well known facts from Fourier analysis. To prove the estimate (25) we use multiple partial integration. Finally, we can prove the other statements by the standard technique of the circle method (see Chapter 11 of [14], for example).

□

We can now define

N_{0} = N_{0}(N) = P ∫_{t_{0}P}^{P} H(1 − \frac{x^{2}}{P^{2}} ) \frac{dx}{log x} , (26)

where t_{0} = (1 − t_{2})^{1/2} ∈ (0, 1) and t_{2} is specified in Lemma 11. Obviously

N_{0} ≍ \frac{P^{2}}{log P} . (27)

3 Propositions which imply Theorem 1.

3.1 Statement of the propositions.

Define

Ω_{d}(n) = ∑_{\begin{matrix}x_{1}^{2}+x_{2}^{2}+x_{3}^{2}=n \\ x_{i}≡0 (d_{i}) \end{matrix}} ω(x). (28)

A formal application of the circle method suggests that the sum Ω_{d}(n) can be approximated (at least in some average sence) by the expression

M_{d,Q}(n) = \frac{P H(nN^{−1})}{d_{1}d_{2}d_{3}} ∑_{q≤Q} q^{−3} ∑_{a(q)^{∗}} e_{q}(−an) S_{d}(q, a) . (29)

It is clear that M_{d,Q}(n) is always real. Denote G_{d,Q}(n) = Ω_{d}(n) − M_{d,Q}(n) (30) and consider the sum

E(D, Q) = ∑_{(D)} τ(d_{1})τ(d_{2})τ(d_{3}) ∑_{k≤P} |G_{d,Q}(N − k^{2})| , (31)

where ∑_{(D)} means that the summation is taken over squarefree odd integers d_{1}, d_{2}, d_{3} ≤ D.

It is easy to see that E(D, Q) ≪ P^{2+ε} for a large range of D and Q. This simple estimate, however, is useless for our aims. To prove a non–trivial result we need an estimate of the shape (33) for a suitable Q and for D as large as possible. This is the most difficult part of the paper.

The following proposition holds: Proposition 1. Suppose that Q = P^{20/23} , D = P^{2/69 − 10ε} . (32) Then we have E(D, Q) ≪ P^{2−ε}. (33)

This statement cannot be applied directly to our problem because of the complicated form of the main term. We will state another proposition, which is suitable for applying the sieve method.

Define

L_{d}(N) = ∑_{\begin{matrix}p^{2}+x_{1}^{2}+x_{2}^{2}+x_{3}^{2}=N \\ x_{i}≡0 (d_{i}) \end{matrix}} ω(x) . (34)

Suppose that β_{i}(d), i = 1, 2, 3, are real functions satisfying β_{i}(d) = 0 if μ(d) = 0 or 2 | d (35) and |β_{i}(d)| ≤ τ(d) . (36) Let

H(D) = ∑_{(D)} β_{1}(d_{1})β_{2}(d_{2})β_{3}(d_{3}) ( L_{d}(N) − \frac{N_{0}(N) Σ_{0}(d, N)}{d_{1}d_{2}d_{3}} ) , (37)

where Σ_{0}, N_{0} and L_{d}(N) are defined by (16), (26) and (34), respectively.

Proposition 2. If D = P^{2/69−10ε} then we have

H(D) ≪ P^{2}(log P)^{−A} . (38)

At the beginning of the proof of Proposition 1 we will impose some simple restrictions on Q and D only. We will impose more severe restrictions later and we will explain the final choice (32) at the end of the proof.

For M_{d,Q}(n) to be a good approximation to Ω_{d}(n) we have to take Q sufficiently large. However, it becomes difficult to work with M_{d,Q}(n) if Q is too large. Now we assume only that

Q ≤ P^{1−ε} . (39)

The reason for introducing this restriction is that in this case we are in a position to apply Lemma 1 with x ≍ P. Another reason is that the exponential sum W_{d,Q}(α), defined by (52), has a comparetively simple behaviour. This becomes clear from Lemma 16.

For D we assume that

D = P^{α_{0}} where α_{0} ∈ (0, 1) . (40)

We note that estimates of the shape (33) and (38) with any small fixed α_{0} imply a nontrivial result for our additive problem. The result becomes better if α_{0} is larger, so our aim is to establish (33) and (38) with α_{0} as large as possible.

3.2 Beginning of the proof of Proposition 1.

Consider the sum E(D, Q), defined by (31). We apply Cauchy’s inequality to get

E^{2}(D, Q) ≪ (log P)^{12} P E_{0}, (41)

where

E_{0} = ∑_{(D)} d_{1}d_{2}d_{3} ∑_{k≤P} | G_{d,Q}(N − k^{2}) |^{2} .

To estimate this sum we apply the “square sieve”, developed by the first author in [7]. We take

R = P^{α_{1}} where α_{1} ∈ (0, 1) . (42)

We shall specify the value of the constant α_{1} later. Consider the quantity

κ(n, R) = (\frac{log R}{R} ∑_{\begin{matrix}R<p≤2R \\ p∤N \end{matrix}} (\frac{N − n}{p}) )^{2} .

If n = N − k^{2} for some integer k ∈ (0, P] then κ(n, R) ≫ 1 and, obviously, κ(n, R) ≥ 0 for all n. Therefore we have

E_{0} ≪ ∑_{(D)} d_{1}d_{2}d_{3} ∑_{n∈ℤ} κ(n, R) | G_{d,Q}(n) |^{2} .

We use the definition of κ(n, R) to get

E_{0} ≪ ( \frac{log R}{R} )^{2} |E_{1}| + \frac{log R}{R} E_{2}, (43)

where

(44)  E_{1} = ∑_{(D)} d_{1}d_{2}d_{3} ∑_{(R)} ∑_{n∈ℤ} (\frac{N − n}{pp′}) | G_{d,Q}(n) |^{2} ,

(45)  E_{2} = ∑_{(D)} d_{1}d_{2}d_{3} ∑_{n∈ℤ} | G_{d,Q}(n) |^{2} .

From this point onwards ∑_{(R)} means that the summation is taken over primes p and p′ such that R < p, p′ ≤ 2R, (pp′, N) = 1 and p ≠ p′.

3.3 The estimation of the sum E_{2}.

3.3.1 Preparation.

We use (30) and (45) to represent E_{2} in the form

E_{2} = E_{2}^{(1)} − 2E_{2}^{(2)} + E_{2}^{(3)} , (46)

where E_{2}^{(i)}, i = 1, 2, 3, come from the consecutive terms in the expansion |G|^{2} = Ω^{2} − 2 Ω M + M^{2} .

First we shall prove that

E_{2}^{(i)} = ∑_{(D)} d_{1}d_{2}d_{3} J_{2}^{(i)}, i = 1, 2, 3, (47)

where

(48)  J_{2}^{(1)} = ∫_{0}^{1} |f_{d}(α)|^{2}dα,
(49)  J_{2}^{(2)} = ∫_{0}^{1} f_{d}(α)W_{d,Q}(−α)dα,
(50)  J_{2}^{(3)} = ∫_{0}^{1} |W_{d,Q}(α)|^{2}dα

and where

(51)  f_{d}(α) = ∏_{i=1}^{3} f_{d_{i}}(α),  f_{d}(α) = ∑_{\begin{matrix}x∈ℤ \\ x≡0 (d) \end{matrix}} ω(x) e(αx^{2}),

(52)  W_{d,Q}(α) = ∑_{n∈ℤ} M_{d,Q}(n) e(αn).

The identity (47) for E_{2}^{(1)} is a consequence of the equalities

∑_{n∈ℤ} Ω_{d}^{2}(n) = ∑_{\begin{matrix}x_{1}^{2}+x_{2}^{2}+x_{3}^{2}=y_{1}^{2}+y_{2}^{2}+y_{3}^{2} \\ x_{i},y_{i}≡0 (d_{i}) \end{matrix}} ω(x)ω(y)

= ∑_{\begin{matrix}x,y∈ℤ^{3} \\ x_{i},y_{i}≡0 (d_{i}) \end{matrix}} ω(x)ω(y) ∫_{0}^{1} e(α(x_{1}^{2} + x_{2}^{2} + x_{3}^{2} − y_{1}^{2} − y_{2}^{2} − y_{3}^{2}))dα

= J_{2}^{(1)} .

To prove the identity (47) for E_{2}^{(2)} we notice that

∑_{n∈ℤ} Ω_{d}(n)M_{d,Q}(n)

= ∑_{\begin{matrix}n∈ℤ, x∈ℤ^{3} \\ x_{i}≡0 (d_{i}) \end{matrix}} ω(x)M_{d,Q}(n) ∫_{0}^{1} e(α(x_{1}^{2} + x_{2}^{2} + x_{3}^{2} − n))dα = J_{2}^{(2)}.

Finally, the identity (47) for E_{2}^{(3)} is a consequence of

∑_{n∈ℤ} M_{d,Q}^{2}(n) = ∫_{0}^{1} |W_{d,Q}(α)|^{2}dα

3.3.2 An asymptotic formula for E_{2}^{(1)}.

Consider the integral J_{2}^{(1)} defined by (48). We study it by means of the Kloosterman method [15]. We consider the Farey dissection of order P for the unit interval and find that

J_{2}^{(1)} = ∑_{q≤P} ∑_{a(q)^{∗}} ∫_{B(q,a)} |f_{d}(\frac{a}{q} + β)|^{2}dβ, (53)

where

B(q, a) = [−(q(q + q′))^{−1}, (q(q + q′′))^{−1}] (54) and where q′ and q′′ are defined by

P < q + q′, q + q′′ ≤ q + P,  q + q′ ≡ \overline{a} (q),  q + q′′ ≡ −\overline{a} (q). (55)

We shall find an expression for the integrand in the right hand side of (53) in which the variables a and β are separated. The following Lemma holds:

Lemma 12. Suppose that q, d, b ∈ ℕ , h ∈ ℤ , β ∈ ℝ , q ≤ P , |β| ≤ (qP)^{−1} and d, b ≤ P ^{c} for some constant c > 0 . Then for any constant γ ≥ 1 we have

f_{d}(\frac{h}{b} + β) = \frac{P}{bd} ∑_{|n|≤γ b d q^{−1} P^{ε}} S(b, hd^{2}, n) I(βN, −\frac{P n}{bd}) + O(P^{−A}) .

The constant in the O – term depends on c, A and ε .

Proof: Using (51) we get

f_{d}(\frac{h}{b} + β) = ∑_{y∈ℤ} ω(dy) e_{b}(h d^{2} y^{2}) e(β d^{2} y^{2}) = ∑_{m (b)} e_{b}(h d^{2} m^{2}) Z_{m} (56)

where

Z_{m} = ∑_{\begin{matrix}y∈ℤ \\ y≡m (b) \end{matrix}} ω(dy) e(β d^{2} y^{2}) .

Consider Z_{m}. We apply Poisson’s formula and use (18) and (19) to get

(57)  Z_{m} = ∑_{x∈ℤ} ω( d (m + bx) ) e( β d^{2} (m + bx)^{2} )

= ∑_{n∈ℤ} ∫_{−∞}^{∞} ω( d (m + bx) ) e( β d^{2} (m + bx)^{2} ) e(−nx) dx

= \frac{P}{bd} ∑_{n∈ℤ} e_{b}(nm) I(β N, −\frac{nP}{bd} ) .

The absolute convergence of the last series is an easy consequence of Lemma 9 (i). Indeed, if n ≠ 0 then we have

I(β N, −\frac{nP}{bd} ) ≪ ( 1 + |β|^{k} N^{k} ) (\frac{|n| P}{bd} )^{−k} ≪ (\frac{bd}{|n|q} )^{k} . (58)

Here k can be arbitrarily large integer and the constant in ≪ – symbol depends on k.

We use (6), (56) and (57) to get

f_{d}(\frac{h}{b} + β) = \frac{P}{bd} ∑_{n∈ℤ} S(b, h d^{2}, n) I(β N, −\frac{nP}{bd} ) .

It remains to estimate the contribution X arising from the terms with |n| > γbdq^{−1}P^{ε}. We take k = [ε^{−1}(2c + A + 3)] and apply (6) and (58). We use the elementary inequality ∑_{n>α} n^{−k} ≤ α^{−k} + α^{−k+1}, which holds true for α > 0 and for k ≥ 2. We conclude that X ≪ P^{−A} and this proves the Lemma.

□

We use (8), (19), (20) and Lemmas 2 (ii) and 12 to find that the integrand in the right hand side of (53) equals

(59)  \frac{P^{6}}{q^{6}(d_{1}d_{2}d_{3})^{2}} ∑_{n∈N_{d,q}} ∑_{l∈N_{d,q}} S_{d}(q, a, n)S_{d}(q, −a, −l)
I_{d}(βN, −P q^{−1}n)I_{d}(−βN, P q^{−1}l) + O(P^{−A}),

where we have set

(60) N_{d,q}(H) = {n ∈ ℤ^{3} : |n_{i}| ≤ 4d_{i}HP^{ε}, n_{i} ≡ 0 ((q, d_{i}^{2})), i = 1, 2, 3},
(61)  N_{d,q} = N_{d,q}(1).

We use (47), (53) and (59) and find an expression for E_{2}^{(1)}. Then we change the variable βN = β′ and denote

B′(q, a) = [−N(q(q + q′))^{−1}, N(q(q + q′′))^{−1}]. (62)

We obtain

(63)  E_{2}^{(1)} = P^{4} ∑_{(D)}(d_{1}d_{2}d_{3})^{−1} ∑_{q≤P} q^{−6} ∑_{n∈N_{d,q}} ∑_{l∈N_{d,q}}

× ∑_{a(q)^{∗}} S_{d}(q, a, n)S_{d}(q, −a, −l)T_{d}(n, l, q, a) + O(P^{−A}),

where

T_{d}(n, l, q, a) = ∫_{B′(q,a)} I_{d}(β, −P q^{−1}n)I_{d}(−β, P q^{−1}l)dβ. (64)

Define

(65) λ(d, n, l, H) = \begin{cases}∑_{i=1}^{3} d_{i}^{−2} ( n_{i}^{2} H^{−2} + l_{i}^{2} ) & if (n, l) ≠ (0, 0) , \\ 1 & otherwise , \end{cases}

and let

λ(d, n, l) = λ(d, n, l, 1) . (66)

We appeal to (23), (55), (62), Lemma 9 (iii) and Lemma 10 (i) to get

(67) T_{d}(n, l, q, a) = \begin{cases}O(q^{2}P^{−2+2ε}λ(d, n, l)^{−1}) & if (n, l) ≠ (0, 0) , \\ κ_{0} + O(q^{2}P^{−2}) & otherwise . \end{cases}

From Lemma 2 (ii) and (iii) we find that

∑_{a(q)^{∗}} |S_{d}(q, a, n)S_{d}(q, −a, −l)| ≪ q^{4}ξ(q, d) , (68)

where

ξ(q, d) = ∏_{i=1}^{3}(q, d_{i}^{2}) . (69)

We use (63), (67) – (69) to obtain E_{2}^{(1)} = U_{2} + O(P^{2+2ε}Y(P)), (70) where

U_{2} = κ_{0}P^{4} ∑_{(D)}(d_{1}d_{2}d_{3})^{−1} ∑_{q≤P} q^{−6} ∑_{a(q)^{∗}} |S_{d}(q, a)|^{2} (71)

and

(72) Y(K) = ∑_{q≤K} ∑_{(D)} ∑_{n∈N_{d,q}} ∑_{l∈N_{d,q}} \frac{ξ(q, d)}{d_{1}d_{2}d_{3}λ(d, n, l)}.

It remains to estimate Y. We first prove two simple results Lemma 13. Suppose that d_{1}, d_{2}, d_{3} ≤ D . Then we have

∑_{l∈N_{d,1}} λ(d, 0, l)^{−1} ≪ D^{3}P^{ε} .

Proof: The sum under consideration is ≪ 1 + T_{1} + T_{2} + T_{3} , where T_{ν} is the contribution from l such that |l_{ν}|d_{ν}^{−1} = max_{i=1,2,3} ( |l_{i}|d_{i}^{−1} ) > 0 . Consider, for example, T_{1} . We have

T_{1} ≪ ∑_{1≤l_{1}≤d_{1}P^{ε}} \frac{d_{1}^{2}}{l_{1}^{2}} (1 + \frac{d_{2}l_{1}}{d_{1}} ) (1 + \frac{d_{3}l_{1}}{d_{1}} ) ≪ D^{3}P^{ε} .

This proves the Lemma.

□

Lemma 14. Let q ≤ P . The following estimate holds:

∑_{d≤D} \frac{(q, d^{2})}{d} ≪ DP^{ε} .

Proof: We have

∑_{d≤D} \frac{(q, d^{2})}{d} ≤ ∑_{d≤D} \frac{(q, d)^{2}}{d} ≤ ∑_{d≤D}(q, d) ≪ DP^{ε} .

□

We are now in a position to estimate Y.

Lemma 15. Suppose that K ≤ P. For the sum Y(K), defined by (72), we have

Y(K) ≪ KD^{6}P^{7ε}.

Proof: It is clear that Y(K) ≪ Y′ + Y′′ , where Y′ is the contribution of the terms with l = 0 and Y′′ is the contribution of the other terms.

Consider Y′′ . Using Lemma 13 we get

∑_{\begin{matrix}l∈N_{d,q} \\ l≠0 \end{matrix}} λ(d, n, l )^{−1} ≪ ∑_{l∈N_{d,q}} λ(d, 0, l )^{−1} ≪ D^{3}P^{ε}

and, obviously,

∑_{n∈N_{d,q}} 1 ≪ P^{3ε} ∏_{i=1}^{3} (1 + \frac{d_{i}}{(q, d_{i}^{2})}) .

Now we use (69) to get

Y′′ ≪ D^{3}P^{4ε} ∑_{q≤K} ( ∑_{d≤D} \frac{(q, d^{2})}{d}(1 + \frac{d}{(q, d^{2})}) )^{3} .

To estimate the sum over d we apply Lemma 14 and find that Y′′ ≪ KD^{6}P^{7ε}.

Consider now Y′ . We apply Lemma 13 with rôles of n and l reversed and get ∑_{n∈N_{d,q}} λ(d, n, 0)^{−1} ≪ D^{3}P^{ε}. Hence

Y′ ≪ D^{3}P^{ε} ∑_{q≤K} ( ∑_{d≤D} \frac{(q, d^{2})}{d} )^{3} .

Now we use Lemma 14 again and obtain Y′ ≪ KD^{6}P^{4ε} . This proves the Lemma.

□

From (70) and Lemma 15 we obtain

E_{2}^{(1)} = U_{2} + O(P^{3+9ε}D^{6}). (73)

3.3.3 An asymptotic formula for E_{2}^{(2)}.

Consider the integral J_{2}^{(2)}, defined by (49). Using the Farey decomposition of order P of the unit interval we get

J_{2}^{(2)} = ∑_{q≤P} ∑_{a(q)^{∗}} ∫_{B(q,a)} f_{d}(\frac{a}{q} + β) W_{d,Q}( − \frac{a}{q} − β)dβ . (74)

As in the previous section, we first separate the terms involving a and β. For f_{d} we apply Lemma 12. To deal with W_{d,Q} we use the following

Lemma 16. Suppose that Q ≤ P^{1−ε}, |β| ≤ (qP)^{−1} and (a, q) = 1. Then we have

W_{d,Q}(\frac{a}{q} + β) = \begin{cases}\frac{P^{3}}{q^{3} d_{1}d_{2}d_{3}} S_{d}(q, a) I^{3}(βN) + O(P^{−A}) & if q ≤ Q , \\ O(P^{−A}) & if Q < q ≤ P . \end{cases}

Proof: We use definitions (29) and (52) of M_{d,Q} and W_{d,Q}, respectively, to get

W_{d,Q}(\frac{a}{q} + β) = \frac{P}{d_{1}d_{2}d_{3}} ∑_{q_{1}≤Q} q_{1}^{−3} ∑_{a_{1}(q_{1})^{∗}} S_{d}(q_{1}, a_{1}) Φ ,

where

Φ = ∑_{n∈ℤ} H(\frac{n}{N} ) e( (\frac{a}{q} − \frac{a_{1}}{q_{1}} + β)n ) .

We apply the Poisson formula, change the variable in the integrals and then apply Lemma 11:

Φ = N ∑_{m∈ℤ} ∫_{−∞}^{∞} H(x) e( (\frac{a}{q} − \frac{a_{1}}{q_{1}} + β − m)Nx ) dx

= N ∑_{m∈ℤ} I^{3}( (\frac{a}{q} − \frac{a_{1}}{q_{1}} + β − m)N ) .

We may suppose that 0 ≤ a ≤ q − 1 and 0 ≤ a_{1} ≤ q_{1} − 1 . In this case we have

|\frac{a}{q} − \frac{a_{1}}{q_{1}} + β| ≤ 1 − \frac{1}{2P} .

We apply Lemma 9 (ii) and easily find that the contribution to Φ coming from m ≠ 0 is O(P^{−A}) .

Consider the term corresponding to m = 0 . Suppose that a/q ≠ a_{1}/q_{1}. We have

|(\frac{a}{q} − \frac{a_{1}}{q_{1}} + β)N| ≥ \frac{|aq_{1} − a_{1}q|}{qq_{1}} N − |β| N ≥ \frac{N}{qq_{1}} − \frac{N}{qP} ≥ \frac{1}{2}P^{ε} .

We use Lemma 9 (ii) again and find that in this case I^{3}( (\frac{a}{q} − \frac{a_{1}}{q_{1}} + β)N ) ≪

P^{−A} .

If Q < q ≤ P then there is no fraction a_{1}/q_{1} equal to a/q.

If q ≤ Q then there is exactly one fraction a_{1}/q_{1} equal to a/q, namely a_{1} = a, q_{1} = q. This proves the Lemma.

□

Using (39) and Lemmas 12 and 16 we find that if q ≤ Q then the integrand in the right–hand side of (74) equals

\frac{P^{6}}{q^{6}(d_{1}d_{2}d_{3})^{2}} ∑_{n∈N_{d,q}} S_{d}(q, a, n)S_{d}(q, −a)
I_{d}(βN, −P q^{−1}n)I^{3}(−βN) + O(P^{−A}).

If Q < q ≤ P then the integrand is O(P^{−A}). Hence, using (47), (64) and (74) we find

E_{2}^{(2)} = P^{4} ∑_{(D)}(d_{1}d_{2}d_{3})^{−1} ∑_{q≤Q} q^{−6} ∑_{n∈N_{d,q}}

× ∑_{a(q)^{∗}} S_{d}(q, a, n)S_{d}(q, −a)T_{d}(n, 0, q, a) + O(P^{−A}).

Now we use (67) – (69) and Lemma 15 to get

E_{2}^{(2)} = U_{2}^{∗} + O(P^{3+9ε}D^{6}), (75)

where

U_{2}^{∗} = κ_{0}P^{4} ∑_{(D)}(d_{1}d_{2}d_{3})^{−1} ∑_{q≤Q} q^{−6} ∑_{a(q)^{∗}} |S_{d}(q, a)|^{2}. (76)

Let us compare U_{2}^{∗} with U_{2}, defined by (71). We use (68), (69), (71), (76) and Lemma 14 to get

(77)  U_{2} − U_{2}^{∗} ≪ P^{4} ∑_{(D)}(d_{1}d_{2}d_{3})^{−1} ∑_{Q<q≤P} q^{−6} ∑_{a(q)^{∗}} |S_{d}(q, a)|^{2}

≪ P^{4} ∑_{(D)}(d_{1}d_{2}d_{3})^{−1} ∑_{Q<q≤P} q^{−2}ξ(q, d)

≪ P^{4} ∑_{Q<q≤P} q^{−2}( ∑_{d≤D} \frac{(q, d^{2})}{d})^{3}

≪ P^{4+3ε}D^{3} ∑_{Q<q} q^{−2} ≪ P^{4+3ε}D^{3}Q^{−1}.

Formulas (75) and (77) imply

E_{2}^{(2)} = U_{2} + O((P^{3}D^{6} + P^{4}D^{3}Q^{−1})P^{9ε}) . (78)

3.3.4 An asymptotic formula for E_{2}^{(3)}.

For the integral J_{2}^{(3)}, defined by (50), we have

J_{2}^{(3)} = ∑_{q≤P} ∑_{a(q)^{∗}} ∫_{B(q,a)} |W_{d,Q}(\frac{a}{q} + β)|^{2}dβ. (79)

Using (39), (47), (79) and Lemma 16 we obtain

(80)  E_{2}^{(3)} = P^{6} ∑_{(D)}(d_{1}d_{2}d_{3})^{−1} ∑_{q≤Q} q^{−6}

× ∑_{a(q)^{∗}} |S_{d}(q, a)|^{2} ∫_{B(q,a)} |I(βN)|^{6}dβ + O(P^{−A}) .

Now we change the variable in the integral and use (64) and (67). Then we proceed as in the previous section to get

(81)  E_{2}^{(3)} = U_{2}^{∗} + O(P^{3+9ε}D^{6})
= U_{2} + O((P^{3}D^{6} + P^{4}D^{3}Q^{−1})P^{9ε}) .

3.3.5 The estimate for E_{2}.

From (46), (73), (78) and (81) we obtain

E_{2} ≪ (P^{3}D^{6} + P^{4}D^{3}Q^{−1})P^{9ε} . (82)

Consider the inequalities (41), (43) and (82). It is clear that in order to obtain (33) we must impose further restrictions on D, Q and R. From this point onwards we assume that

D ≤ R^{1/6}P^{−10ε} , P^{1+20ε}D^{3}R^{−1} ≤ Q . (83)

3.4 The estimation of the sum E_{1}.

3.4.1 Preparation. Consider the sum E_{1}, defined by (44). We represent it in the form E_{1} = E_{1}^{(1)} − 2E_{1}^{(2)} + E_{1}^{(3)}, (84)

where E_{1}^{(i)} are the contributions of the consecutive terms from the expansion |G|^{2} = Ω^{2} − 2ΩM + M^{2}.

First we shall prove that for the sums E_{1}^{(i)} the following formulas hold:

E_{1}^{(i)} = ∑_{(R)} \frac{γ(pp′)}{pp′} ∑_{(D)} d_{1}d_{2}d_{3} ∑_{s(pp′)^{∗}} (\frac{s}{pp′} ) e_{pp′}(−sN) J_{1}^{(i)}, i = 1, 2, 3, (85)

where γ(pp′) is the Gauss sum,defined by (5), and

(86)  J_{1}^{(1)} = ∫_{0}^{1} f_{d}(α + \frac{s}{pp′} )f_{d}( − α) dα ,
(87)  J_{1}^{(2)} = ∫_{0}^{1} f_{d}(α + \frac{s}{pp′} )W_{d}( − α) dα ,
(88)  J_{1}^{(3)} = ∑_{n∈ℤ} M_{d,Q}^{2}(n) e_{pp′}(sn) .

Consider the sum E_{1}^{(1)} . We have

(89)  E_{1}^{(1)} = ∑_{(R)} ∑_{(D)} d_{1}d_{2}d_{3} ∑_{n∈ℤ} (\frac{N − n}{pp′}) Ω_{d}^{2}(n)

= ∑_{(R)} ∑_{(D)} d_{1}d_{2}d_{3} Γ_{1}^{(1)},

say. Obviously, (28) implies

Γ_{1}^{(1)} = ∑_{\begin{matrix}x_{1}^{2}+x_{2}^{2}+x_{3}^{2}=y_{1}^{2}+y_{2}^{2}+y_{3}^{2} \\ x_{i},y_{i}≡0 (d_{i}) \end{matrix}} ω(x) ω(y) (\frac{N − (x_{1}^{2} + x_{2}^{2} + x_{3}^{2})}{pp′}) .

Using (51) and taking into account the elementary properties of the Gauss sum γ(q) we get

(90)  Γ_{1}^{(1)} = ∑_{l(pp′)} (\frac{l}{pp′} ) ∑_{\begin{matrix}x_{1}^{2}+x_{2}^{2}+x_{3}^{2}=y_{1}^{2}+y_{2}^{2}+y_{3}^{2} \\ x_{1}^{2}+x_{2}^{2}+x_{3}^{2}≡N−l (pp′) \\ x_{i},y_{i}≡0 (d_{i}) \end{matrix}} ω(x) ω(y)

= ∑_{l(pp′)} (\frac{l}{pp′} ) ∑_{\begin{matrix}x,y∈ℤ^{3} \\ x_{i},y_{i}≡0 (d_{i}) \end{matrix}} ω(x) ω(y)

× ∫_{0}^{1} e(α(x_{1}^{2} + x_{2}^{2} + x_{3}^{2} − y_{1}^{2} − y_{2}^{2} − y_{3}^{2}) ) dα

× \frac{1}{pp′} ∑_{s(pp′)} e_{pp′}(s(x_{1}^{2} + x_{2}^{2} + x_{3}^{2} − N + l) )

= \frac{1}{pp′} ∑_{l(pp′)} (\frac{l}{pp′} ) ∑_{s(pp′)} e_{pp′}(s(l − N)) J_{1}^{(1)}

= \frac{γ(pp′)}{pp′} ∑_{s(pp′)^{∗}} (\frac{s}{pp′} ) e_{pp′}(−sN) J_{1}^{(1)} ,

where J_{1}^{(1)} is defined by (86). From (89) and (90) we obtain formula (85) for E_{1}^{(1)}.

Consider E_{1}^{(2)}. We have

(91)  E_{1}^{(2)} = ∑_{(R)} ∑_{(D)} d_{1}d_{2}d_{3} ∑_{n∈ℤ} (\frac{N − n}{pp′}) Ω_{d}(n) M_{d,Q}(n)

= ∑_{(R)} ∑_{(D)} d_{1}d_{2}d_{3} Γ_{1}^{(2)} ,

say. Furthermore

Γ_{1}^{(2)} = ∑_{l(pp′)} (\frac{l}{pp′} ) ∑_{\begin{matrix}n∈ℤ \\ n≡N−l (pp′) \end{matrix}} Ω_{d}(n) M_{d,Q}(n) = ∑_{l(pp′)} (\frac{l}{pp′} ) Φ^{∗} , (92)

say. Obviously

Φ^{∗} = ∑_{\begin{matrix}n∈ℤ, x∈ℤ^{3} \\ x_{1}^{2}+x_{2}^{2}+x_{3}^{2}=n≡N−l (pp′) \\ x_{i}≡0 (d_{i}) \end{matrix}} ω(x) M_{d,Q}(n) .

By the previous formula and (51), (52) we obtain

Φ^{∗} = ∑_{\begin{matrix}n∈ℤ, x∈ℤ^{3} \\ x_{i}≡0 (d_{i}) \end{matrix}} ω(x) M_{d,Q}(n) ∫_{0}^{1} e(α(x_{1}^{2} + x_{2}^{2} + x_{3}^{2} − n)) dα

× \frac{1}{pp′} ∑_{s(pp′)} e_{pp′}(s(x_{1}^{2} + x_{2}^{2} + x_{3}^{2} − N + l))

= \frac{1}{pp′} ∑_{s(pp′)} e_{pp′}(s(l − N)) J_{1}^{(2)} ,

where J_{1}^{(2)} is defined by (87). It remains to substitute this expression for Φ^{∗} in (92) and to change the order of summation over l and s. The Gauss sum γ(q) appears again. We use (91) and obtain formula (85) for E_{1}^{(2)}.

Consider E_{1}^{(3)}. We have

(93)  E_{1}^{(3)} = ∑_{(R)} ∑_{(D)} d_{1}d_{2}d_{3} ∑_{n∈ℤ} (\frac{N − n}{pp′}) M_{d,Q}^{2}(n)

= ∑_{(R)} ∑_{(D)} d_{1}d_{2}d_{3} Γ_{1}^{(3)} ,

say. It is clear that

(94)  Γ_{1}^{(3)} = ∑_{l(pp′)} (\frac{l}{pp′} ) ∑_{\begin{matrix}n∈ℤ \\ n≡N−l (pp′) \end{matrix}} M_{d,Q}^{2}(n)

= ∑_{l(pp′)} (\frac{l}{pp′} ) ∑_{n∈ℤ} M_{d,Q}^{2}(n)\frac{1}{pp′} ∑_{s(pp′)} e_{pp′}(s(n − N + l))

= \frac{γ(pp′)}{pp′} ∑_{s(pp′)^{∗}} (\frac{s}{pp′} )e_{pp′}(−sN) J_{1}^{(3)} ,

where J_{1}^{(3)} is specified by (88). From (93) and (94) we obtain formula (85) for E_{1}^{(3)} .

3.4.2 The sum E_{1}^{(1)} .

In this section we will establish the asymptotic formula (109). Its remainder term, however, is still a complicated expression which we shall estimate later.

Consider the integral J_{1}^{(1)}, defined by (86). We represent again the unit interval as an union of Farey arcs

⋃_{q≤P} ⋃_{\begin{matrix}1≤a≤q \\ (a,q)=1 \end{matrix}} (B(q, a) + a/q) ,

where B(q, a) is defined by (54). We decompose further the set of integration according to the size of (app′ + sq, qpp′) . We note that (app′ + sq, qpp′) | (pp′)^{2} and find

(95) J_{1}^{(1)} = ∑_{δ|(pp′)^{2}} ∑_{q≤P}

× ∑_{\begin{matrix}a(q)^{∗} \\ (app′+sq,qpp′)=δ \end{matrix}} ∫_{B(q,a)} f_{d}(\frac{a}{q} + \frac{s}{pp′} + β) f_{d}( − \frac{a}{q} − β) dβ .

We use Lemmas 2 (ii) and 12 to see that the integrand in the last formula equals

\frac{P^{6}δ^{3}}{q^{6}(pp′)^{3}(d_{1}d_{2}d_{3})^{2}} ∑_{n∈N_{d,q}(R^{2}δ^{−1})} ∑_{l∈N_{d,q}} (96)

× S_{d}(\frac{qpp′}{δ} , \frac{app′ + sq}{δ} , n) S_{d}(q, −a, −l)

× I_{d}(βN, −\frac{P δ}{qpp′} n ) I_{d}( − βN, \frac{P}{q} l ) + O(P^{−A}) ,

where the sets N_{d,q}(H) and N_{d,q} are defined by (60) and (61), respectively.

Some care is needed with the modulus of the congruence in the definition of N_{d,q}(H). According to (83) we have d_{i} ≤ D < R^{1/6} < p, p′. Thus, since δ | (pp′)^{2}, we have

(qpp′δ^{−1}, (app′ + sq)δ^{−1}d_{i}^{2}) = (qpp′δ^{−1}, d_{i}^{2}) = (q, d_{i}^{2}).

It follows that the modulus of the congruence occuring in the definition of N_{d,q}(H) may be taken to be (q, d_{i}^{2}) instead of (qpp′δ^{−1}, d_{i}^{2}).

We use (85), (95) and take into account the expression (96) for the integrand. In this way we find a formula for E_{1}^{(1)} . Then we change the variable βN = β′ and use (62) to obtain

E_{1}^{(1)} = P^{4} ∑_{(R)} \frac{γ(pp′)}{(pp′)^{4}} ∑_{(D)} ∑_{δ|(pp′)^{2}} ∑_{q≤P} \frac{δ^{3}}{d_{1} d_{2} d_{3} q^{6}}

× ∑_{n∈N_{d,q}(R^{2}δ^{−1})} ∑_{l∈N_{d,q}} ∑_{s(pp′)^{∗}} (\frac{s}{pp′} ) e_{pp′}(−sN) ∑_{\begin{matrix}a(q)^{∗} \\ (app′+sq, qpp′)=δ \end{matrix}}

× ∫_{B′(q,a)} S_{d}(\frac{qpp′}{δ} , \frac{app′ + sq}{δ} , n) S_{d}(q, −a, −l)

× I_{d}(β, −\frac{P δ}{qpp′} n) I_{d}( − β, \frac{P}{q} l) dβ + O(P^{−A}) .

We decompose E_{1}^{(1)} as follows:

E_{1}^{(1)} = U + V + O(P^{−A}) . (97)

Here U is the contribution of the terms with n = l = 0 and V is the contribution of the other terms.

Consider V. We change the order of summation over a and integration over β. Using (62) we conclude that in the new expression for V the domain of integration is {|β| ≤ P q^{−1}} and we sum over a satisfying the previous conditions and such that \overline{a} belongs to a set A(q, β) of residues (mod q) . There is no restriction on \overline{a} if |β| ≤ P (2q)^{−1} . For the other β the set A(q, β) is not necessarily a complete set of residues modulo q, but we have

∑_{\begin{matrix}a(q)^{∗} \\ (app′+sq, qpp′)=δ \\ \overline{a}∈A(q,β) \end{matrix}} … = ∑_{−q/2<v≤q/2} σ(v, q, β) ∑_{\begin{matrix}a(q)^{∗} \\ (app′+sq, qpp′)=δ \end{matrix}} e_{q}(\overline{a}v) …, (98)

where the function σ satisfies

|σ(v, q, β)| ≪ (1 + |v|)^{−1} . (99)

Detailed explanation of this technique can be found in Section 3 of [8], for example.

Lemma 10 (i) and (65) imply that if (n, l) ≠ (0, 0) then

∫_{−∞}^{∞} |I_{d}(β, −\frac{P δ}{qpp′} n) I_{d}( − β, \frac{P}{q} l) | dβ ≪ \frac{q^{2} P^{2ε−2}}{λ(d, n, l, R^{2}δ^{−1})} . (100)

We use (98) – (100) and Lemma 6 (i) to get

V = O(V^{∗}) , (101)

where

(102) V^{∗} = P^{2+2ε}R^{−7} ∑_{(R)} ∑_{(D)} ∑_{δ|(pp′)^{2}} ∑_{q≤P} \frac{δ^{3}}{d_{1} d_{2} d_{3} q^{4}}

× ∑_{n∈N_{d,q}(R^{2}δ^{−1})} ∑_{l∈N_{d,q}} ∑_{|v|≤P} \frac{|W (d, n, l, p, p′, q, v, δ)|}{(1 + |v|) λ(d, n, l, R^{2}δ^{−1})}

and

W (d, n, l, p, p′, q, v, δ) = ∑_{s(pp′)^{∗}} (\frac{s}{pp′} ) e_{pp′}(−sN) (103)

× ∑_{\begin{matrix}a(q)^{∗} \\ (app′+sq,qpp′)=δ \end{matrix}} e_{q}(\overline{a}v) S_{d}(\frac{qpp′}{δ} , \frac{app′ + sq}{δ} , n) S_{d}(q, −a, −l) .

Consider U. We represent it as

U = U′ + U′′ , (104)

where in U′ the integration is taken over the subset {|β| ≤ P (2q)^{−1}} of B′(q, a) and, respectively, in U′′ we integrate over B′(q, a) ∩ {|β| > P (2q)^{−1}} .

Consider first U′′ . We change the order of summation over a and integration and proceed as in the treatment of V . We apply the simple estimate

∫_{|β|>P (2q)^{−1}} |I(β)|^{6}dβ ≪ q^{2}P^{−2} (105)

and use (65) to establish that

U′′ = O(V^{∗}) . (106)

Consider now U′. In this case the set of integration does not depend on a. We extend the domain of integration to the real line. Taking into account (23),

(65) and (105) we find that

U = U_{1} + O(V^{∗}) , (107)

where

(108) U_{1} = κ_{0} P^{4} ∑_{(R)} \frac{γ(pp′)}{(pp′)^{4}} ∑_{(D)} ∑_{δ|(pp′)^{2}} ∑_{q≤P} \frac{δ^{3} W (d, 0, 0, p, p′, q, 0, δ)}{d_{1} d_{2} d_{3} q^{6}} .

From (97), (101), (104), (106) and (107) we obtain

E_{1}^{(1)} = U_{1} + O(V^{∗}) . (109)

3.4.3 The sum E_{1}^{(2)} .

In this section we shall establish the asymptotic formula (111). The main term in it coinsides with the main term in (109). The remainder terms are complicated expressions and we shall study them later.

Consider the integral J_{1}^{(2)}, defined by (87). We decompose again the unit interval as in the previous section and find that

J_{1}^{(2)} = ∑_{δ|(pp′)^{2}} ∑_{q≤P} (110)

× ∑_{\begin{matrix}a(q)^{∗} \\ (app′+sq,qpp′)=δ \end{matrix}} ∫_{B(q,a)} f_{d}(\frac{a}{q} + \frac{s}{pp′} + β) W_{d,Q}( − \frac{a}{q} − β) dα .

To deal with f_{d} and W_{d,Q} we use Lemmas 12 and 16, respectively. We conclude that if q ≤ Q then the integrand in the right hand side of (110) equals

\frac{P^{6}δ^{3}}{q^{6}(pp′)^{3}(d_{1}d_{2}d_{3})^{2}} ∑_{n∈N_{d,q}(R^{2}δ^{−1})} S_{d}(\frac{qpp′}{δ} , \frac{app′ + sq}{δ} , n) S_{d}(q, −a)

× I_{d}(βN, −\frac{P δ}{qpp′} n ) I^{3}( − βN) + O(P^{−A}) .

If Q < q ≤ P then the integrand is O(P^{−A}) . From this observation, (85) and (110) we find an expression for E_{1}^{(2)}. We change the variable in the integral and obtain

E_{1}^{(2)} = P^{4} ∑_{(R)} \frac{γ(pp′)}{(pp′)^{4}} ∑_{(D)} ∑_{δ|(pp′)^{2}} ∑_{q≤Q} \frac{δ^{3}}{d_{1} d_{2} d_{3} q^{6}}

× ∑_{n∈N_{d,q}(R^{2}δ^{−1})} ∑_{s(pp′)^{∗}} (\frac{s}{pp′} ) e_{pp′}(−sN) ∑_{\begin{matrix}a(q)^{∗} \\ (app′+sq, qpp′)=δ \end{matrix}}

× ∫_{B′(q,a)} S_{d}(\frac{qpp′}{δ} , \frac{app′ + sq}{δ} , n) S_{d}(q, −a)

× I_{d}(β, −\frac{P δ}{qpp′} n) I^{3}( − β) dβ + O(P^{−A}) .

Proceeding as in Section 3.4.2 we find that the contribution to E_{1}^{(2)} from the terms with n ≠ 0 is O(V^{∗}) , where V^{∗} is defined by (102).

Let U^{∗} be the contribution to E_{1}^{(2)} from the terms with n = 0 . Arguing as in Section 3.4.2 we see that U^{∗} = U^{∗∗} + O(V^{∗}) , where in U^{∗∗} the integration is taken over the real line.

Let us compare the expression U^{∗∗} with U_{1}, defined by (108). The only difference is that in the first one we sum over q ≤ Q , whilst in the second the summation is taken over q ≤ P . Hence we have

E_{1}^{(2)} = U_{1} + O(V^{∗}) + O(D^{∗}) , (111)

where

(112)  D^{∗} = P^{4}R^{−7} ∑_{(R)} ∑_{(D)} ∑_{δ|(pp′)^{2}}

× ∑ ; min (Q , Qδ(pp′)^{−1})<q≤P \frac{δ^{3} |W (d, 0, 0, p, p′, q, 0, δ)|}{d_{1} d_{2} d_{3} q^{6}} ,

and where W is defined by (103). Here we have defined D^{∗} with a longer range of summation for q than is needed at this point. We do this because we shall encounter, in the next section, an error term whose estimation will involve the sum D^{∗} as defined above.

3.4.4 The sum E_{1}^{(3)} .

The object of this section is to establish the asymptotic formula (121). The main term there coincides with the main terms in (109) and (111). The error terms in (121) are complicated and we shall estimate them in the next sections.

Consider the quantity J_{1}^{(3)}, defined by (88). We use (29) to get

(113) J_{1}^{(3)} = \frac{P^{2}}{(d_{1}d_{2}d_{3})^{2}} ∑_{q_{1},q_{2}≤Q}(q_{1}q_{2})^{−3}

× ∑_{a_{1}(q_{1})^{∗}, a_{2}(q_{2})^{∗}} S_{d}(q_{2}, −a_{2}) S_{d}(q_{1}, −a_{1}) B(\frac{a_{1}}{q_{1}} + \frac{s}{pp′} + \frac{a_{2}}{q_{2}} ),

where

B(α) = ∑_{n∈ℤ} H^{2}(\frac{n}{N} ) e(nα) (114)

We apply Poisson’s summation formula to obtain

B(α) = P^{2} ∑_{k∈ℤ} ∫_{−∞}^{∞} H^{2}(t) e( (α − k)Nt )dt .

Now we use Lemma 11 and after some standard calculations we find that

B(α) = \begin{cases}κ_{0}P^{2} + O(P^{−A}) & if α ∈ ℤ, \\ O(P^{−A}) & if ||α|| ≥ P^{ε−2}. \end{cases} (115)

Using (85) and (113) we find an expression for E_{1}^{(3)}. Then we take into account (115) to obtain

E_{1}^{(3)} = D + E + O(P^{−A}), (116)

where D is the contribution of the terms with \frac{a_{1}}{q_{1}} + \frac{s}{pp′} + \frac{a_{2}}{q_{2}} ∈ ℤ and E comes from the terms with 0 < ||\frac{a_{1}}{q_{1}} + \frac{s}{pp′} + \frac{a_{2}}{q_{2}} || < P^{ε−2}.

Consider E. We have

E = P^{2} ∑_{(R)} \frac{γ(pp′)}{pp′} ∑_{(D)}(d_{1}d_{2}d_{3})^{−1} ∑_{n∈ℤ} H^{2}(\frac{n}{N} ) F,

where

F = ∑_{q_{1},q_{2}≤Q}(q_{1}q_{2})^{−3} ∑_{s(pp′)^{∗}} (\frac{s}{pp′} )e_{pp′}(−sN) (117)

× ∑_{\begin{matrix}a_{1}(q_{1})^{∗} , a_{2}(q_{2})^{∗} \\ 0<|| \frac{a_{1}}{q_{1}} + \frac{s}{pp′} + \frac{a_{2}}{q_{2}} ||<P^{ε−2} \end{matrix}} S_{d}(q_{1}, −a_{1})S_{d}(q_{2}, −a_{2})e(n(\frac{a_{1}}{q_{1}} + \frac{s}{pp′} + \frac{a_{2}}{q_{2}} )).

Therefore we have E = O(E^{∗}) , (118) where

E^{∗} = P^{2}R^{−1} ∑_{(R)} ∑_{(D)}(d_{1}d_{2}d_{3})^{−1} ∑_{0<n<N} |F|, (119)

Consider D. If we denote (a_{1}pp′ + sq_{1}, q_{1}pp′) = δ then it is easy to see that δ | (pp′)^{2}. The condition \frac{a_{1}}{q_{1}} + \frac{s}{pp′} + \frac{a_{2}}{q_{2}} ∈ ℤ implies that q_{2} = q_{1}pp′δ^{−1} and a_{2} ≡ −(a_{1}pp′ + sq_{1})δ^{−1} (q_{2}) and consequently

S_{d}(q_{2}, −a_{2}) = S_{d}(\frac{q_{1}pp′}{δ} , \frac{a_{1}pp′ + sq_{1}}{δ} ) .

We use this observation and (115) to get

D = κ_{0}P^{4} ∑_{(R)} \frac{γ(pp′)}{(pp′)^{4}} ∑_{(D)} ∑_{δ|(pp′)^{2}}

× ∑ ; q≤min (Q , Qδ(pp′)^{−1}) \frac{δ^{3} W (d, 0, 0, p, p′, q, 0, δ)}{d_{1} d_{2} d_{3} q^{6}} + O(P^{−A}) .

It is now clear that D = U_{1} + O(D^{∗}) , (120) where U_{1} and D^{∗} are defined by (108) and (112), respectively.

From (116), (118) and (120) we obtain

E_{1}^{(3)} = U_{1} + O(D^{∗}) + O(E^{∗}) . (121) 3.4.5 The estimation of V^{∗} and D^{∗}. We use definitions (102) and (112) of V^{∗} and D^{∗}, respectively, to get

(122)  V^{∗} ≪ V_{1} + V_{p} + V_{p^{2}} + V_{pp′} + V_{p^{2}p′} + V_{p^{2}p′^{2}} ,
(123)  D^{∗} ≪ D_{1} + D_{p} + D_{p^{2}} + D_{pp′} + D_{p^{2}p′} + D_{p^{2}p′^{2}} ,

where V_{δ} and D_{δ} are the contributions coming from the corresponding values of δ .

The estimations of V_{1} and V_{p} are the most difficult because the domain of summation over n is largest in these cases. The other terms from the right hand sides of (122) and (123) are much simpler.

Two more Lemmas. Consider the function

(124) Θ(d, n, l, q, h, v) = ∑_{a(q)^{∗}} e_{q}(\overline{a} v) S_{d}(q, ah^{2}, n) S_{d}(q, −a, −l) .

To emphasize the dependence of q we will write also Θ(q) for simplicity. Define

η = η(d, n, l, h, v) = (2hd_{1}d_{2}d_{3})^{2} v + ∑_{i=1}^{3} \frac{(d_{1}d_{2}d_{3})^{2}}{d_{i}^{2}} ( h^{2}l_{i}^{2} − n_{i}^{2} ) . (125)

The following Lemma holds:

Lemma 17. The function Θ is multiplicative with respect to q. Suppose that d_{1}, d_{2}, d_{3} are squarefree odd numbers. If (h, 2) = 1 then Θ(2^{α}) ≪ (2^{α})^{4}. If (q, 2h) = 1 then

|Θ(q)| ≤ q^{3} ( q ξ(q, d) , η(d, n, l, h, v) ) . (126)

Proof: First we note that if (q_{1}, q_{2}) = (a_{1}, q_{1}) = (a_{2}, q_{2}) = 1 then the following simple identity holds:

e_{q_{1}q_{2}}(\overline{(a_{1}q_{2} + a_{2}q_{1})} m) = e_{q_{1}}(\overline{(a_{1}q_{2}^{2})} m)e_{q_{2}}(\overline{(a_{2}q_{1}^{2})} m) . (127)

The multiplicativity of Θ with respect to q is an easy consequence of (127) and Lemma 2 (i). If (h, 2) = 1 then the estimate Θ(2^{α}) ≪ (2^{α})^{4} follows from Lemma 2 (iii).

It remains to prove (126) provided that (q, 2h) = 1. It is enough to establish that for any prime p ∤ 2h and for any integer k we have

|Θ(p^{k})| ≤ p^{3k} ( p^{k} ξ(p^{k}, d) , η ) . (128)

We may suppose that n_{i}, l_{i} ≡ 0 ((p^{k}, d_{i}^{2})) because otherwise, according to Lemma 2 (ii), we have Θ = 0 and (128) is true.

Let d_{i} = p^{μ_{i}}e_{i} , where p ∤ e_{i} . Then μ_{i} = 0 or μ_{i} = 1 and (p^{k}, d_{i}^{2}) = p^{ν_{i}} , where ν_{i} = min(k, 2μ_{i}) . Let n′_{i} = n_{i} p^{−ν_{i}} and l′_{i} = l_{i} p^{−ν_{i}} . We have by Lemma 2 (ii) and (iv)

Θ(p^{k}) = ∏_{i=1}^{3} ( (\frac{−1}{p^{k−ν_{i}}} ) p^{2ν_{i}} S^{2}(p^{k−ν_{i}}, 1)) ρ(p^{k}) ,

where

ρ(p^{k}) = ∑_{a(p^{k})^{∗}} e_{p^{k}}(\overline{a} v) ∏_{\begin{matrix}i=1 \\ ν_{i}<k \end{matrix}}^{3} e_{p^{k−ν_{i}}} (\overline{(4ae_{i}^{2})}_{p^{k−ν_{i}}} (l′_{i}^{2} − \overline{(h^{2})}_{p^{k−ν_{i}}} n′_{i}^{2}) ) .

Applying Lemma 2 (iii) we get |Θ(p^{k})| ≤ p^{3k+ν_{1}+ν_{2}+ν_{3}} |ρ(p^{k})| . (129)

Consider ρ(p^{k}) . It is easy to see that if p ∤ A then e_{p^{k}}( \overline{(A)}_{p^{k}} p^{ν}) = e_{p^{k−ν}} ( \overline{(A)}_{p^{k−ν}} ) . Hence

ρ(p^{k}) = ∑_{a(p^{k})^{∗}} e_{p^{k}}(\overline{a} v + ∑_{\begin{matrix}i=1 \\ ν_{i}<k \end{matrix}}^{3} p^{ν_{i}} \overline{(4ae_{i}^{2})}_{p^{k}} (l′_{i}^{2} − \overline{(h^{2})}_{p^{k}} n′_{i}^{2}) ) .

We can already drop the condition ν_{i} < k from the domain of summation of the inner sum because the terms with ν_{i} = k do not contribute to ρ(p^{k}) . We obtain

ρ(p^{k}) = ∑_{a(p^{k})^{∗}} e_{p^{k}}(\overline{(a (2he_{1}e_{2}e_{3})^{2})} η′ ) = c_{p^{k}}(η′) ,

where c_{q}(n) is the Ramanujan sum, defined by (4), and

η′ = (2he_{1}e_{2}e_{3})^{2} v + ∑_{i=1}^{3} p^{ν_{i}} (e_{1}e_{2}e_{3})^{2} e_{i}^{−2} ( h^{2}l′_{i}^{2} − n′_{i}^{2} ) .

Lemma 5 gives |ρ(p^{k})| ≤ (p^{k}, η′) . (130) Therefore using (129) and (130) we get

|Θ(p^{k})| ≤ p^{3k+ν_{1}+ν_{2}+ν_{3}} (p^{k}, η′) = p^{3k}(p^{k+ν_{1}+ν_{2}+ν_{3}}, η′′) ,

where η′′ = p^{ν_{1}+ν_{2}+ν_{3}} η′ .

If k ≥ 2 then ν_{i} = 2μ_{i} and η′′ = η , so the inequality (128) holds.

Consider the case k = 1 . Now ν_{i} = μ_{i} and we have |Θ(p)| ≤ p^{3} (p^{1+μ_{1}+μ_{2}+μ_{3}}, η′′) ≤ p^{3} (p^{1+μ_{1}+μ_{2}+μ_{3}}, η′′′) , (131) where

η′′′ = p^{μ_{1}+μ_{2}+μ_{3}} η′′ = (2hd_{1}d_{2}d_{3})^{2} v + ∑_{i=1}^{3} p^{μ_{i}} (d_{1}d_{2}d_{3})^{2} d_{i}^{−2} ( h^{2}l_{i}^{2} − n_{i}^{2} ) .

It is easy to see that p^{1+μ_{1}+μ_{2}+μ_{3}} | (η′′′ − η) , whence (p^{1+μ_{1}+μ_{2}+μ_{3}}, η′′′) = (p^{1+μ_{1}+μ_{2}+μ_{3}}, η) . (132) Using (131) and (132) we conclude that (128) is true for k = 1 as well. The Lemma is proved.

□

Lemma 18. Suppose that H is an integer such that R ≤ H ≤ 4R^{2} and let

(133)  D(H) = ∑_{d_{1},d_{2},d_{3}≤D} ∑_{q≤P HR^{−2}} ∑_{|v|≤P}

× ∑_{n∈N_{d,q}(H)} ∑_{l∈N_{d,q}} \frac{( q ξ(q, d) , η(d, n, l, H, v) )}{q d_{1} d_{2} d_{3} (1 + |v|) λ(d, n, l, H)} ,

The following estimate holds:

D(H) ≪ (P H^{2}R^{−2} + H^{3})D^{6}P^{10ε} , (134)

Proof: We have

D(H) ≪ D′ + D′′ , (135)

where D′ is the contribution of the terms for which η(d, n, l, H, v) = 0 , and D′′ is the contribution of the other terms.

Consider D′′ . In this case the quantity η^{∗} = ξ(q, d)^{−1} η is a non–zero integer because n and l satisfy the congruence conditions imposed in the definitions (60) and (61) of N_{d,q}(H) and N_{d,q}, respectively. It is easy to see that ∑_{q≤P HR^{−2}} (q, η^{∗}) q^{−1} ≪ P^{ε} . Hence, using (69), we get

(136) D′′ ≪ ∑_{d_{1},d_{2},d_{3}≤D} ∑_{\begin{matrix}h_{i}|d_{i}^{2} \\ i=1,2,3 \end{matrix}} \frac{h_{1}h_{2}h_{3}}{d_{1}d_{2}d_{3}}

× ∑_{n∈N^{∗}} ∑_{l∈N_{d,1}} λ(d, n, l, H)^{−1} ∑_{\begin{matrix}|v|≤P \\ η≠0 \end{matrix}} \frac{1}{1 + |v|} ∑_{\begin{matrix}q≤P HR^{−2} \\ (q,d_{i}^{2})=h_{i} \\ i=1,2,3 \end{matrix}} \frac{(q, η^{∗})}{q}

≪ P^{2ε} H ,

where

H = ∑_{d_{1},d_{2},d_{3}≤D} ∑_{\begin{matrix}h_{i}|d_{i}^{2} \\ i=1,2,3 \end{matrix}} \frac{h_{1}h_{2}h_{3}}{d_{1}d_{2}d_{3}} ∑_{n∈N^{∗}} ∑_{l∈N_{d,1}} λ(d, n, l, H)^{−1}

and

N^{∗} = { n ∈ ℤ^{3} : |n_{i}| ≤ 4d_{i}HP^{ε} , n_{i} ≡ 0 (h_{i}) , i = 1, 2, 3 } .

We have

H ≪ H′ + H′′ (137) where H′ is the contribution of the terms with l = 0 and H′′ is the contribution of the other terms.

Consider H′′. According to (65) and Lemma 13 we have

∑_{\begin{matrix}l∈N_{d,1} \\ l≠0 \end{matrix}} λ(d, n, l, H)^{−1} ≪ ∑_{l∈N_{d,1}} λ(d, 0, l)^{−1} ≪ D^{3} P^{ε} .

Our assumption (83) implies that D ≤ R^{1/6} and we also have R ≤ H. Thus h_{i} ≤ d_{i}^{2} ≤ d_{i}D ≤ d_{i}H, and we easily get

∑_{n∈N^{∗}} 1 ≪ H^{3} P^{3ε} \frac{d_{1}d_{2}d_{3}}{h_{1}h_{2}h_{3}} .

Therefore

H′′ ≪ H^{3} D^{6} P^{7ε} (138)

Consider H′ . First we estimate the quantity

B = ∑_{n∈N^{∗}} λ(d, n, 0, H)^{−1} .

We have

B ≪ 1 + B_{1} + B_{2} + B_{3} ,

where B_{ν} comes from the terms with |n_{ν}|d_{ν}^{−1} = max_{i=1,2,3} (|n_{i}|d_{i}^{−1}) > 0 . Consider, for example, B_{1} . Using again the inequalities D ≤ R^{1/6} and R ≤ H we obtain

B_{1} ≪ H^{2} ∑_{\begin{matrix}1≤n_{1}≤4d_{1}HP^{ε} \\ n_{1}≡0 (h_{1}) \end{matrix}} \frac{d_{1}^{2}}{n_{1}^{2}} (1 + \frac{d_{2} n_{1}}{h_{2} d_{1}} ) (1 + \frac{d_{3} n_{1}}{h_{3} d_{1}} ) ≪ H^{3} P^{ε} \frac{d_{1}d_{2}d_{3}}{h_{1}h_{2}h_{3}} .

Obviously, the same estimate holds for B as well. Therefore

H′ = ∑_{d_{1},d_{2},d_{3}≤D} ∑_{\begin{matrix}h_{i}|d_{i}^{2} \\ i=1,2,3 \end{matrix}} \frac{h_{1}h_{2}h_{3}}{d_{1}d_{2}d_{3}} B ≪ H^{3} D^{3} P^{2ε} . (139)

The estimates (136) – (139) imply

D′′ ≪ H^{3} D^{6} P^{9ε} . (140)

Now consider D′ . We have

D′ ≪ ∑_{d_{1},d_{2},d_{3}≤D} ∑_{q≤P HR^{−2}} \frac{ξ(q, d)}{d_{1}d_{2}d_{3}} X , (141)

X = ∑_{|v|≤P} ∑_{l∈N_{d,q}} ∑_{\begin{matrix}n∈N_{d,q}(H) \\ η=0 \end{matrix}} ( (1 + |v|) λ(d, n, l, H) )^{−1} .

We divide X into two parts:

X = X′ + X′′ , (142)

where X′ is the contribution from the terms with l = 0 and X′′ comes from the other terms.

Consider X′′ . Using (65) we get

X′′ ≪ ∑_{l∈N_{d,q}} λ(d, 0, l)^{−1} ∑_{|v|≤P} \frac{K(d, l, q, H, v)}{(1 + |v|)}

where K is the number of n ∈ N_{d,q}(H) such that η(d, n, l, H, v) = 0 .

Let us estimate K . There are O( H P^{ε} d_{1} (q, d_{1}^{2})^{−1} ) admissible values of n_{1} . If d, l, H, v, n_{1} are fixed then the equation η = 0 determines at most O(P^{ε}) pairs n_{2}, n_{3} . Therefore

K(d, l, q, H, v) ≪ H P^{2ε} d_{1} (q, d_{1}^{2})^{−1}

and, according to Lemma 13,

X′′ ≪ P^{4ε} D^{3} H d_{1} (q, d_{1}^{2})^{−1} . (143)

Consider X′ . The contribution to X′ coming from n = 0 is O(P^{ε}). Using (65) and (125) we conclude that if n ≠ 0 and η(d, n, 0, H, v) = 0 then we have v > 0 and λ(d, n, 0, H)^{−1} ≪ v^{−1} . Therefore

X′ ≪ P^{ε} + ∑_{1≤v≤P} v^{−2} K(d, 0, q, H, v) ≪ H P^{2ε} d_{1} (q, d_{1}^{2})^{−1} . (144)

From (141) – (144) and Lemma 14 we obtain

D′ ≪ P^{1+6ε}H^{2}D^{6}R^{−2} . (145)

The estimate (134) is a consequence of (135), (140) and (145).

This completes the proof of the Lemma.

□

Consider V_{1} and D_{1}. We use (102), (112) and the definitions of V_{1} and D_{1}, given at the beginning of this section. We note that if (a, q) = (s, pp′) = 1 then the conditions (q, pp′) = 1 and (app′ + sq, qpp′) = 1 are equivalent. We impose the first of these conditions in the domain of summation over q and omit the second from the domain of summation over a. We obtain

(146)  V_{1} ≪ P^{2+2ε} R^{−7} ∑_{(R)} ∑_{(D)} (d_{1}d_{2}d_{3})^{−1} ∑_{\begin{matrix}q≤P \\ (q,pp′)=1 \end{matrix}} q^{−4}

× ∑_{n∈N_{d,q}(R^{2})} ∑_{l∈N_{d,q}} ∑_{|v|≤P} \frac{|W_{1}(d, n, l, p, p′, q, v)|}{(1 + |v|) λ(d, n, l, R^{2})}

and

D_{1} ≪ P^{4} R^{−7} ∑_{(R)} ∑_{(D)} ∑_{\begin{matrix}Q(pp′)^{−1}<q≤P \\ (q,pp′)=1 \end{matrix}} \frac{|W_{1}(d, 0, 0, p, p′, q, 0)|}{q^{6} d_{1}d_{2}d_{3}} , (147)

where

W_{1}(d, n, l, p, p′, q, v) = ∑_{s(pp′)^{∗}} (\frac{s}{pp′} ) e_{pp′}(−sN)

× ∑_{a(q)^{∗}} e_{q}(\overline{a}v)S_{d}(qpp′, app′ + sq, n) S_{d}(q, −a, −l) .

We apply Lemma 2 (i) and (iv) and find that

(148)  S(qpp′, (app′ + sq)d_{i}^{2}, n_{i}) = S(q, a(pp′)^{2}d_{i}^{2}, n_{i}) S(pp′, sq^{2}d_{i}^{2}, n_{i})

= S(q, a(pp′)^{2}d_{i}^{2}, n_{i}) S(pp′, 1) (\frac{s}{pp′} ) e_{pp′}( − \overline{(4sq^{2}d_{i}^{2})} n_{i}^{2}) .

Therefore, using (4), (8), (124), (148) and Lemmas 2 (iii) and 4 we find

(149) W_{1}(d, n, l, p, p′, q, v) = S^{3}(pp′, 1) K(pp′, −N, −\overline{(4q^{2})} ∑_{i=1}^{3} \overline{(d_{i}^{2})} n_{i}^{2} )
× Θ(d, n, l, q, pp′, v)
≪ R^{4} |Θ(d, n, l, q, pp′, v) | .

We represent each q ≤ P, satisfying (q, pp′) = 1, in the form q = 2^{α}t, where (t, 2pp′) = 1. Hence, using Lemma 17, we get

(150) Θ(d, n, l, q, pp′, v) ≪ (2^{α})^{4} t^{3} (tξ(t, d) , η(d, n, l, pp′, v)) ,

where ξ and η are defined by (69) and (125), respectively.

Obviously (q, d_{i}^{2}) = (t, d_{i}^{2}), hence using (60) and (61) we get

N_{d,q}(R^{2}) = N_{d,t}(R^{2}) , N_{d,q} = N_{d,t} . (151)

Formulas (149) – (151) imply

∑_{\begin{matrix}q≤P \\ (q,pp′)=1 \end{matrix}} ∑_{n∈N_{d,q}(R^{2})} ∑_{l∈N_{d,q}} \frac{|W_{1}(d, n, l, p, p′, q, v)|}{q^{4} λ(d, n, l, R^{2})} (152)

≪ R^{4} ∑_{2^{α}≤P} ∑_{\begin{matrix}t≤P 2^{−α} \\ (t,2pp′)=1 \end{matrix}} ∑_{n∈N_{d,t}(R^{2})} ∑_{l∈N_{d,t}} \frac{(t ξ(t, d) , η(d, n, l, pp′, v))}{t λ(d, n, l, R^{2})}

≪ R^{4}P^{ε} ∑_{\begin{matrix}q≤P \\ (q,pp′)=1 \end{matrix}} ∑_{n∈N_{d,q}(R^{2})} ∑_{l∈N_{d,q}} \frac{(q ξ(q, d) , η(d, n, l, pp′, v))}{q λ(d, n, l, R^{2})} .

From (133), (146) and (152) we obtain

V_{1} ≪ P^{2+3ε}R^{−3} ∑_{(R)} D(pp′) .

Now we apply Lemma 18 to get

V_{1} ≪ (P^{3}R + P^{2}R^{5})D^{6}P^{15ε} . (153)

The estimation of D_{1} is much easier. We use (124), (149) and Lemma 2 (ii) and (iii) to get

W_{1}(d, 0, 0, p, p′, q, 0) ≪ R^{4} q^{4} ξ(q, d) .

We substitute this estimate for W_{1} in (147). Then we apply Lemma 14 and after some calculations we obtain

D_{1} ≪ P^{4+5ε}Q^{−1}D^{3}R . (154)

Actually, a stronger estimate for D_{1} is available because in this particular case the Kloosterman sum reduces to Ramanujan’s sum. This improvement, however, would not have any influence on our final result.

Consider V_{p} and D_{p}. We use (102), (112) and the definitions of V_{p} and D_{p}. If (app′ +sq, qpp′) = p then p | q and p′ ∤ q , so we impose these conditions in the range of summation over q . As p′ ∤ app′ + sq and (ap′ + s\frac{q}{p}, \frac{q}{p}) = 1 we can relax the condition on a to (ap′ + s\frac{q}{p}, p) = 1 . We put q = rp and get

(155)  V_{p} ≪ P^{2+2ε} R^{−8} ∑_{(R)} ∑_{(D)}(d_{1}d_{2}d_{3})^{−1} ∑_{\begin{matrix}r≤P R^{−1} \\ (r,p′)=1 \end{matrix}} r^{−4}

× ∑_{n∈N_{d,r}(R)} ∑_{l∈N_{d,r}} ∑_{|v|≤P} \frac{|W_{2}(d, n, l, p, p′, r, v)|}{(1 + |v|) λ(d, n, l, R)}

and

D_{p} ≪ P^{4} R^{−10} ∑_{(R)} ∑_{(D)} ∑_{\begin{matrix}Q(pp′)^{−1}<r≤P R^{−1} \\ (r,p′)=1 \end{matrix}} \frac{|W_{2}(d, 0, 0, p, p′, r, 0)|}{r^{6} d_{1}d_{2}d_{3}} , (156)

where

W_{2}(d, n, l, p, p′, r, v) = ∑_{s(pp′)^{∗}} (\frac{s}{pp′} ) e_{pp′}(−sN)

× ∑_{\begin{matrix}a(rp)^{∗} \\ (ap′+sr,p)=1 \end{matrix}} e_{rp}(\overline{a}v) S_{d}(rpp′, ap′ + sr, n) S_{d}(rp, −a, −l) .

We represent each r ≤ P R^{−1} satisfying (r, p′) = 1 in the form r = r_{0}r_{1}t , (157) where r_{0} = 2^{α} , r_{1} = p^{β} , (t, 2pp′) = 1 . (158) We note that N_{d,r}(R) = N_{d,t}(R) , N_{d,r} = N_{d,t} . (159)

It is clear that the set { cp′ + c′p | c(p)^{∗}, c′(p′)^{∗} } (160) is a reduced system of residues modulo pp′. Similarly, the set

{ a_{0} r_{1}p t + a_{1} r_{0} t + b r_{0} r_{1}p | a_{0}(r_{0})^{∗}, a_{1}(r_{1}p)^{∗}, b(t)^{∗} }

is a reduced system of residues modulo r_{0} r_{1}p t = rp.

It is easy to see that if s = cp′ + c′p and a = a_{0} r_{1}p t + a_{1} r_{0} t + b r_{0} r_{1}p then the condition (ap′ + sr, p) = 1 is equivalent to (a_{1} + cr_{1}, p) = 1. From these observations, (127) and the elementary properties of the Jacobi symbol we obtain

W_{2}(d, n, l, p, p′, r_{0}r_{1}t, v) = (\frac{p′}{p} )(\frac{p}{p′} ) (161)

× ∑_{c(p)^{∗}} ∑_{c′(p′)^{∗}} (\frac{c}{p}) (\frac{c′}{p′} ) e_{p}(−cN) e_{p′}(−c′N)

× ∑_{a_{0}(r_{0})^{∗}} ∑_{\begin{matrix}a_{1}(r_{1}p)^{∗} \\ (a_{1}+cr_{1},p)=1 \end{matrix}} ∑_{b(t)^{∗}}

× e_{r_{0}}( \overline{a_{0}(r_{1}p t)^{2}} v) e_{r_{1}p}( \overline{a_{1}(r_{0} t)^{2}} v) e_{t}( \overline{b(r_{0} r_{1}p)^{2}} v) S′ S′′ , where

S′ = S_{d}(r_{0} r_{1}p t p′ , (a_{0} r_{1}p t + a_{1} r_{0} t + b r_{0} r_{1}p) p′ + (cp′ + c′p) r_{0} r_{1} t , n) and

S′′ = S_{d}(r_{0} r_{1}p t , −(a_{0} r_{1}p t + a_{1} r_{0} t + b r_{0} r_{1}p) , −l) .

We use (8) and Lemma 2 (i) to get

(162)  S′ = S_{d}(p′, c′(r_{0} r_{1}p t)^{2}, n)
×S_{d}(r_{0}, a_{0}(r_{1}p t p′)^{2}, n)
×S_{d}(r_{1}p, (a_{1} + cr_{1}) (r_{0} t p′)^{2}, n)
×S_{d}(t, b(r_{0} r_{1}p p′)^{2}, n)

and

(163)  S′′ = S_{d}(r_{0}, −a_{0}(r_{1}p t)^{2}, −l)
×S_{d}(r_{1}p, −a_{1}(r_{0} t)^{2}, −l)
×S_{d}(t, −b(r_{0} r_{1}p)^{2}, −l) .

From (124) and (161) – (163) we get

W_{2}(d, n, l, p, p′, r_{0}r_{1}t, v) = (\frac{p′}{p} ) (\frac{p}{p′} ) W′_{2} W_{2}^{(0)} W_{2}^{(1)} W_{2}^{(2)} , (164)

where

W′_{2} = ∑_{c′(p′)^{∗}} (\frac{c′}{p′} ) e_{p′}(−c′N) S_{d}(p′, c′(r_{0} r_{1}p t)^{2}, n) ,
W_{2}^{(0)} = Θ(d, n, l, r_{0}, p′, v) .
W_{2}^{(1)} = ∑_{c(p)^{∗}} (\frac{c}{p}) e_{p}(−cN) ∑_{\begin{matrix}a_{1}(r_{1}p)^{∗} \\ (a_{1}+cr_{1},p)=1 \end{matrix}} e_{r_{1}p}(\overline{a_{1} (r_{0} t)^{2}}v)
× S_{d}(r_{1}p, (a_{1} + cr_{1})(r_{0} t p′)^{2}, n ) S_{d}(r_{1}p, −a_{1}(r_{0} t)^{2}, −l ) ,
W_{2}^{(2)} = Θ(d, n, l, t, p′, v) .

From (4), (8) and Lemmas 2 (iii), (iv) and 4 we get

W′_{2} = S^{3}(p′, 1) K( p′, −N, − ∑_{i=1}^{3} \overline{4(r_{0} r_{1}p t d_{i})^{2}} n_{i}^{2} ) ≪ R^{2} . (165)

Applying only (8) and Lemma 2 (iii) we find that W_{2}^{(1)} ≪ r_{1}^{4}R^{5} . (166) Finally, Lemma 17 implies W_{2}^{(0)} ≪ r_{0}^{4} (167) and W_{2}^{(2)} ≪ t^{3} (t ξ(t, d), η(d, n, l, p′, v)) . (168)

Taking into account (157) – (159) and (164) – (168) we obtain

∑_{\begin{matrix}r≤P R^{−1} \\ (r,p′)=1 \end{matrix}} ∑_{n∈N_{d,r}(R)} ∑_{l∈N_{d,r}} \frac{|W_{2}(d, n, l, p, p′, r, v)|}{r^{4} λ(d, n, l, R)} (169)

≪ ∑_{2^{α}≤P} ∑_{p^{β}≤P} ∑_{\begin{matrix}t≤P R^{−1} \\ (t,2pp′)=1 \end{matrix}} ∑_{n∈N_{d,t}(R)} ∑_{l∈N_{d,t}}

× \frac{R^{2} (2^{α})^{4} R^{5} (p^{β})^{4} t^{3} (t ξ(t, d), η(d, n, l, p′, v))}{(2^{α} p^{β} t)^{4} λ(d, n, l, R)}

≪ P^{ε}R^{7} ∑_{\begin{matrix}q≤P R^{−1} \\ (q,p′)=1 \end{matrix}} ∑_{n∈N_{d,q}(R)} ∑_{l∈N_{d,q}} \frac{(q ξ(q, d), η(d, n, l, p′, v))}{q λ(d, n, l, R)}

From (155) and (169) we get

V_{p} ≪ P^{2+3ε}R^{−1} ∑_{(R)} D(p′) ,

where D(H) is defined by (133). It remains to apply Lemma 18 and we find V_{p} ≪ (P^{3}R + P^{2}R^{4})D^{6}P^{15ε} . (170)

The estimation of D_{p} is much simpler. For any r satisfying the conditions imposed in (156) we have W_{2}(d, 0, 0, p, p′, r, 0) ≪ R^{7} r^{4} ξ(r, d) . (171) To prove this inequality we first represent r in the form (157), where r_{0}, r_{1}, t satisfy (158). Then we use formula (164), inequalities (165) – (167) as well as the inequality W_{2}^{(2)} ≪ t^{4} ξ(t, d), which is a weak version of (168). The estimate (171) follows.

From (69), (156), (171) and Lemma 14 we obtain D_{p} ≪ P^{4+3ε}Q^{−1}D^{3}R . (172) Here again a sharper estimate is available, but this is of no importance for our result. Consider V_{p^{2}} and D_{p^{2}}. If (app′ + sq, qpp′) = p^{2} then p | q , p^{2} ∤ q , p′ ∤ q . We put q = rp. The condition (app′ + sq, qpp′) = p^{2} is equivalent to ap′ + sr ≡ 0 (p) and, obviously, N_{d,q} = N_{d,r}. We use (102) and (112) to get

(173)  V_{p^{2}} ≪ P^{2+2ε} R^{−5} ∑_{(R)} ∑_{(D)} (d_{1}d_{2}d_{3})^{−1} ∑_{\begin{matrix}r≤P R^{−1} \\ (r,pp′)=1 \end{matrix}} r^{−4}

× ∑_{n∈N_{d,r}} ∑_{l∈N_{d,r}} ∑_{|v|≤P} \frac{|W_{3}(d, n, l, p, p′, r, v)|}{(1 + |v|) λ(d, n, l)}

and

D_{p^{2}} ≪ P^{4} R^{−7} ∑_{(R)} ∑_{(D)} ∑_{\begin{matrix}Q(4R)^{−1}<r≤P R^{−1} \\ (r,pp′)=1 \end{matrix}} \frac{|W_{3}(d, 0, 0, p, p′, r, 0)|}{r^{6} d_{1}d_{2}d_{3}} (174)

where

W_{3}(d, n, l, p, p′, r, v) = ∑_{s(pp′)^{∗}} (\frac{s}{pp′} ) e_{pp′}(−sN)

× ∑_{\begin{matrix}a(rp)^{∗} \\ ap′+sr≡0 (p) \end{matrix}} e_{rp}(\overline{a}v) S_{d}(rp′, (ap′ + sr) p^{−1}, n) S_{d}(rp, −a, −l) .

We use (69) and Lemma 2 (ii) and (iii) to get W_{3}(d, n, l, p, p′, r, v) ≪ r^{4} R^{5} ξ(r, d) . (175) From (72), (173), (175) and Lemma 15 we obtain V_{p^{2}} ≪ P^{2+3ε} R^{2} Y(P R^{−1}) ≪ P^{3+10ε}RD^{6} . (176) To estimate D_{p^{2}} we apply (174), (175) and Lemma 14. We get D_{p^{2}} ≪ P^{4+3ε}Q^{−1}D^{3}R . (177)

Consider V_{pp′} and D_{pp′}. If (app′ + sq, qpp′) = pp′ then pp′ | q . We put q = rpp′ and find

(178)  V_{pp′} ≪ P^{2+2ε} R^{−9} ∑_{(R)} ∑_{(D)} (d_{1}d_{2}d_{3})^{−1} ∑_{r≤P R^{−2}} r^{−4}

× ∑_{n∈N_{d,r}} ∑_{l∈N_{d,r}} ∑_{|v|≤P} \frac{|W_{4}(d, n, l, p, p′, r, v)|}{(1 + |v|) λ(d, n, l )} ,

and

D_{pp′} ≪ P^{4} R^{−13} ∑_{(R)} ∑_{(D)} ∑_{Q(2R)^{−2}<r≤P R^{−2}} \frac{|W_{4}(d, 0, 0, p, p′, r, 0)|}{r^{6} d_{1}d_{2}d_{3}} , (179)

where

W_{4}(d, n, l, p, p′, r, v) = ∑_{s(pp′)^{∗}} (\frac{s}{pp′} ) e_{pp′}(−sN)

× ∑_{\begin{matrix}a(rpp′)^{∗} \\ (a+sr,rpp′)=1 \end{matrix}} e_{rpp′}(\overline{a}v) S_{d}(rpp′, a + sr, n) S_{d}(rpp′, −a, −l) .

Using (69) and Lemma 2 (ii) and (iii) we get W_{4}(d, n, l, p, p′, r, v) ≪ R^{10}r^{4}ξ(r, d) . (180) From (72), (178), (180) and Lemma 15 we get V_{pp′} ≪ P^{2+3ε} R^{3} Y(P R^{−2}) ≪ P^{3+10ε}RD^{6} . (181) Similarly, using (179), (180) and Lemma 14 we obtain D_{pp′} ≪ P^{4+3ε}Q^{−1}D^{3}R . (182)

Consider V_{p^{2}p′} and D_{p^{2}p′}. If (app′ + sq, qpp′) = δ = p^{2}p′ then the conditions |n_{i}| ≤ 4d_{i}R^{2}δ^{−1}P^{ε} imply n = 0 . We have also pp′ | q , p^{2} ∤ q . We put q = rpp′ and find

(183)  V_{p^{2}p′} ≪ P^{2+2ε} R^{−6} ∑_{(R)} ∑_{(D)} (d_{1}d_{2}d_{3})^{−1} ∑_{\begin{matrix}r≤P R^{−2} \\ (r,p)=1 \end{matrix}} r^{−4}

× ∑_{l∈N_{d,r}} ∑_{|v|≤P} \frac{|W_{5}(d, l, p, p′, r, v)|}{(1 + |v|) λ(d, 0, l)}

and

D_{p^{2}p′} ≪ P^{4} R^{−10} ∑_{(R)} ∑_{(D)} ∑_{\begin{matrix}Q(pp′)^{−1}<r≤P R^{−2} \\ (r,p)=1 \end{matrix}} \frac{|W_{5}(d, 0, p, p′, r, 0)|}{r^{6} d_{1}d_{2}d_{3}} , (184)

where

W_{5}(d, l, p, p′, r, v) = ∑_{s(pp′)^{∗}} (\frac{s}{pp′} ) e_{pp′}(−sN)

× ∑_{\begin{matrix}a(rpp′)^{∗} \\ (a+sr,rpp′)=p \end{matrix}} e_{rpp′}(\overline{a}v) S_{d}(rp′, (a + sr)p^{−1}) S_{d}(rpp′, −a, −l) .

We note again that the set (160) is a reduced system of residues modulo pp′. Similarly, the set

{ bp + hrp′ | b(rp′)^{∗}, h(p)^{∗} }

is a reduced system of residues modulo rpp′.

If s = cp′ + c′p and a = bp + hrp′ then the condition (a + sr, rpp′) = p is equivalent to the system of conditions (b + c′r, rp′) = 1 and h + c ≡ 0 (p) and we have

S_{d}(rp′, (a + sr)p^{−1}) = S_{d}(rp′, b + c′r) .

Furthermore, using Lemma 5 we also get

S_{d}(rp′ p, −a, −l) = S_{d}(rp′, −bp^{2}, −l) S_{d}(p, −h(rp′)^{2}, −l) .

From (127) and Lemma 2 (ii) – (iv) we obtain

(185) W_{5}(d, l, p, p′, r, v) = (\frac{p}{p′} )(\frac{p′}{p} ) S^{3}(p, 1)

× K(p, −N, −\overline{(rp′)^{2}}(v + ∑_{i=1}^{3} \overline{(4d_{i}^{2})} l_{i}^{2} ) )

× ∑_{c′(p′)} (\frac{c′}{p′} )e_{p′}(−c′N) ∑_{\begin{matrix}b(rp′)^{∗} \\ (b+c′r,rp′)=1 \end{matrix}} e_{rp′}( \overline{(bp^{2})}v )

× S_{d}(rp′, b + c′r) S_{d}(rp′, −bp^{2}, −l)

≪ R^{7} r^{4} ξ(r, d) .

We apply (183), (185) and Lemmas 13 and 14 to get V_{p^{2}p′} ≪ P^{3+10ε} R D^{6} . (186) Analogously, from (184), (185) and Lemma 14 we obtain D_{p^{2}p′} ≪ P^{4+5ε}Q^{−1}D^{3}R . (187) Consider V_{(pp′)^{2}} and D_{(pp′)^{2}}. If (app′ +sq, qpp′) = δ = (pp′)^{2} then the conditions |n_{i}| ≤ 4d_{i}R^{2}δ^{−1}P^{ε} imply n = 0. We have also pp′ | q , p^{2} ∤ q , p′ ^{2} ∤ q.

We put q = rpp′ and find

(188)  V_{(pp′)^{2}} ≪ P^{2+2ε} R^{−3} ∑_{(R)} ∑_{(D)} (d_{1}d_{2}d_{3})^{−1} ∑_{\begin{matrix}r≤P R^{−2} \\ (r,pp′)=1 \end{matrix}} r^{−4}

× ∑_{l∈N_{d,r}} ∑_{|v|≤P} \frac{|W_{6}(d, l, p, p′, r, v)|}{(1 + |v|) λ(d, 0, l)}

and

D_{(pp′)^{2}} ≪ P^{4} R^{−7} ∑_{(R)} ∑_{(D)} ∑_{\begin{matrix}Q(pp′)^{−1}<r≤P R^{−2} \\ (r,pp′)=1 \end{matrix}} \frac{|W_{6}(d, 0, p, p′, r, 0)|}{r^{6} d_{1}d_{2}d_{3}} , (189)

where

W_{6}(d, l, p, p′, r, v) = ∑_{s(pp′)^{∗}} (\frac{s}{pp′} ) e_{pp′}(−sN)

× ∑_{\begin{matrix}a(rpp′)^{∗} \\ a+sr≡0 (pp′) \end{matrix}} e_{rpp′}(\overline{a}v) S_{d}(r, (a + sr)(pp′)^{−1}) S_{d}(rpp′, −a, −l) .

We note that the set { bpp′ + tr | b(r)^{∗}, t(pp′)^{∗} } is a reduced system of residues modulo rpp′. If a = bpp′ + tr then the condition a + sr ≡ 0 (pp′) is equivalent to t + s ≡ 0 (pp′). We also have (a + sr)(pp′)^{−1} ≡ b (r) and

S_{d}(rpp′, −a, −l) = S_{d}(r, −b(pp′)^{2}, −l) S_{d}(pp′, −tr^{2}, −l) .

We apply (127) and Lemma 2 (ii) – (iv) to find that

(190) W_{6}(d, l, p, p′, r, v) = S^{3}(pp′, 1) K(pp′, −N, −\overline{r^{2}}(v + ∑_{i=1}^{3} \overline{(4d_{i}^{2})} l_{i}^{2} ) )

× ∑_{b(r)^{∗}} e_{r}(\overline{b(pp′)^{2}}v) S_{d}(r, b) S_{d}(r, −b(pp′)^{2}, −l)

≪ R^{4} r^{4} ξ(r, d) .

We use (188), (190) and Lemmas 13 and 14 to find V_{(pp′)^{2}} ≪ P^{3+10ε} R D^{6} . (191) From (189), (190) and Lemma 14 we obtain D_{(pp′)^{2}} ≪ P^{4+5ε}Q^{−1}D^{3}R . (192)

Conclusion. Using (122), (153), (170), (176), (181), (186) and (191) we find that V^{∗} ≪ (P^{3}R + P^{2}R^{5})D^{6}P^{15ε} . (193)

Respectively, using (123), (154), (172), (177), (182), (187) and (192) we obtain D^{∗} ≪ P^{4+5ε}Q^{−1}D^{3}R . (194)

3.4.6 The estimation of E^{∗}.

Consider the sum F, defined by (117). We divide it into parts according to the congruence class of pp′(a_{1}q_{2} + a_{2}q_{1}) + sq_{1}q_{2} modulo q_{1}q_{2}pp′. After some rearrangements we get

(195)  F = ∑_{0<|l|≤4R^{2}Q^{2}P^{ε−2}} ∑_{\begin{matrix}q_{1},q_{2}≤Q \\ |l|(pp′)^{−1}P^{2−ε}<q_{1}q_{2} \end{matrix}}(q_{1}q_{2})^{−3}

× e(\frac{nl}{q_{1}q_{2}pp′} ) ∑_{s(pp′)^{∗}} (\frac{s}{pp′} )e_{pp′}(−sN)

× ∑_{\begin{matrix}a_{1}(q_{1})^{∗} , a_{2}(q_{2})^{∗} \\ pp′(a_{1}q_{2}+a_{2}q_{1})+sq_{1}q_{2}≡l (q_{1}q_{2}pp′) \end{matrix}} S_{d}(q_{1}, −a_{1})S_{d}(q_{2}, −a_{2}) .

From this point onwards we assume that

Q ≤ P^{1−ε}R^{−1/2}. (196)

This inequality implies that in the right hand side of (195) we sum over l satisfying 0 < |l| < R, consequently (l, pp′) = 1 and (q_{1}q_{2}, pp′) = 1. Therefore, the congruence condition in the sum over a_{1}, a_{2} is equivalent to the system of congruences

pp′(a_{1}q_{2} + a_{2}q_{1}) ≡ l (q_{1}q_{2}) , sq_{1}q_{2} ≡ l (pp′).

The second congruence determines s uniquely modulo pp′. Therefore, the sum over s in the right hand side of (195) has exactly one term corresponding to s ≡ l \overline{q_{1}q_{2}} (pp′). Hence we obtain

(197)  F = ∑_{\begin{matrix}0<|l|≤4R^{2}Q^{2}P^{ε−2} \\ (l,pp′)=1 \end{matrix}} ∑_{\begin{matrix}q_{1},q_{2}≤Q \\ |l|(pp′)^{−1}P^{2−ε}<q_{1}q_{2} \\ (q_{1}q_{2},pp′)=1 \end{matrix}}(q_{1}q_{2})^{−3} e(\frac{nl}{q_{1}q_{2}pp′} )

× (\frac{q_{1}q_{2}l}{pp′} ) · e(\frac{−Nl \overline{q_{1}q_{2}}}{pp′} ) · S ,

where

S = ∑_{\begin{matrix}a_{1}(q_{1})^{∗} , a_{2}(q_{2})^{∗} \\ pp′(a_{1}q_{2}+a_{2}q_{1})≡l (q_{1}q_{2}) \end{matrix}} S_{d}(q_{1}, −a_{1})S_{d}(q_{2}, −a_{2}) . (198)

We represent the integers q_{i} as

q_{i} = g_{i}b_{i} , i = 1, 2 , (199)

where

g_{i} = ∏_{\begin{matrix}p^{l}||q_{i} \\ p∤2d_{1}d_{2}d_{3}(q_{1},q_{2}) \end{matrix}} p^{l} ,  i = 1, 2 (200)

and where b_{i} are determined by (199). To produce the set of summation over q_{1}, q_{2} in the right hand side of (197) we have to sum over the set of all quadruples of integers g_{1}, g_{2}, b_{1}, b_{2}, satisfying the conditions

g_{1}b_{1}, g_{2}b_{2} ≤ Q, |l|(pp′)^{−1}P^{2−ε} < b_{1}b_{2}g_{1}g_{2},

(g_{1}g_{2}, 2pp′d_{1}d_{2}d_{3}b_{1}b_{2}) = (g_{1}, g_{2}) = (b_{1}b_{2}, pp′) = 1, (201)

ρ_{d}(b_{1}) = ρ_{d}(b_{2}),

where we have defined

ρ_{d}(m) = ∏_{\begin{matrix}p|m \\ p∤2d_{1}d_{2}d_{3} \end{matrix}} p. (202)

Consider the sum S. Suppose that q_{1}, q_{2} satisfy (199) and (201). We note that the sets { α_{i}b_{i} + β_{i}g_{i} | α_{i}(g_{i})^{∗} , β_{i}(b_{i})^{∗} } , i = 1, 2 are reduced system of residues moduli q_{i}, i = 1, 2. Furthermore, if a_{i} = α_{i}b_{i} + β_{i}g_{i} , i = 1, 2, then the congruence condition, imposed on the domain of summation in (198), is equivalent to the system of the following three conditions

pp′α_{1}b_{1}b_{2}g_{2} ≡ l (g_{1}) , pp′α_{2}b_{1}b_{2}g_{1} ≡ l (g_{2}) , (203)

pp′g_{1}g_{2}(β_{1}b_{2} + β_{2}b_{1}) ≡ l (b_{1}b_{2}) . (204)

We use Lemma 2 (i) and (iv) to find that under conditions (199), (201) we have

(205)  S = ∑_{\begin{matrix}α_{1}(g_{1})^{∗}, α_{2}(g_{2})^{∗} \\ (203) \end{matrix}} S^{3}(g_{1}, −α_{1}) S^{3}(g_{2}, −α_{2}) · S′

= (\frac{−pp′b_{1}b_{2}g_{2}l}{g_{1}} )(\frac{−pp′b_{1}b_{2}g_{1}l}{g_{2}} )S^{3}(g_{1}, 1)S^{3}(g_{2}, 1) · S′ ,

where

S′ = ∑_{\begin{matrix}β_{1}(b_{1})^{∗}, β_{2}(b_{2})^{∗} \\ (204) \end{matrix}} S_{d}(b_{1}, −β_{1}) S_{d}(b_{2}, −β_{2}). (206)

We decompose F into O(log^{2} P) sums F(G_{1}, G_{2}) according to the size of g_{i}. In F(G_{1}, G_{2}) we sum over

g_{i} ∈ (G_{i}, 2G_{i}] , i = 1, 2 . (207)

Consider the sum E^{∗}, defined by (119), and denote by E(G_{1}, G_{2}) the contribution to E^{∗} coming from F(G_{1}, G_{2}). We have

E^{∗} ≪ P^{ε} max_{G_{1},G_{2}≤Q} |E(G_{1}, G_{2})| (208)

and

E(G_{1}, G_{2}) ≪ P^{2}R^{−1} ∑_{(R)} ∑_{(D)}(d_{1}d_{2}d_{3})^{−1} ∑_{0<n<N} |F(G_{1}, G_{2})|. (209)

Consider F(G_{1}, G_{2}). We take into account (197), (199) – (201), (205), (207) and find that

(210) F(G_{1}, G_{2}) = ∑_{\begin{matrix}0<|l|≤4R^{2}Q^{2}P^{ε−2} \\ (l,pp′)=1 \end{matrix}} ∑_{\begin{matrix}g_{1},g_{2},b_{1},b_{2} \\ (201), (207) \end{matrix}}(b_{1}b_{2}g_{1}g_{2})^{−3}

× e(\frac{nl}{g_{1}g_{2}b_{1}b_{2}pp′} ) · (\frac{g_{1}g_{2}b_{1}b_{2}l}{pp′} ) · e(\frac{−Nl\overline{(b_{1}b_{2}g_{1}g_{2})}}{pp′})

× (\frac{−pp′b_{1}b_{2}g_{2}l}{g_{1}} ) · (\frac{−pp′b_{1}b_{2}g_{1}l}{g_{2}} )

× S^{3}(g_{1}, 1) S^{3}(g_{2}, 1) · S′ ,

where S′ is defined by (206) and where the summation over g_{1}, g_{2}, b_{1}, b_{2} is restricted to integers satisfying (201) and (207).

We split further b_{1} = B_{1}Δ, b_{2} = B_{2}Δ, where Δ = (b_{1}, b_{2}). The congruence condition (204) implies Δ | l, so we put l = Δv. Now we can simplify this congruence and we write S′ in the form

S′ = Ξ(pp′g_{1}g_{2}) ,

where

(211)  Ξ(μ) = Ξ(B_{1}, B_{2}, Δ, v, μ)

= ∑_{\begin{matrix}β_{1}(ΔB_{1})^{∗} , β_{2}(ΔB_{2})^{∗} \\ μ(β_{1}B_{2}+β_{2}B_{1})≡v (ΔB_{1}B_{2}) \end{matrix}} S_{d}(ΔB_{1}, −β_{1})S_{d}(ΔB_{2}, −β_{2}) .

We obtain

F(G_{1}, G_{2}) = ∑_{\begin{matrix}0<Δ|v|≤4R^{2}Q^{2}P^{ε−2} \\ (Δv,pp′)=1 \end{matrix}} ∑_{\begin{matrix}g_{1},g_{2},B_{1},B_{2} \\ (213) \end{matrix}} \frac{S^{3}(g_{1}, 1) S^{3}(g_{2}, 1)}{(B_{1}B_{2}Δ^{2}g_{1}g_{2})^{3}} (212)

× e(\frac{nv}{B_{1}B_{2}Δg_{1}g_{2}pp′} ) · (\frac{B_{1}B_{2}g_{1}g_{2}Δv}{pp′} ) · e(\frac{−Nv\overline{(B_{1}B_{2}Δg_{1}g_{2})}}{pp′})

× (\frac{−pp′B_{1}B_{2}g_{2}Δv}{g_{1}} ) · (\frac{−pp′B_{1}B_{2}g_{1}Δv}{g_{2}} ) · Ξ(pp′g_{1}g_{2}) .

In the formula above the summation over g_{1}, g_{2}, B_{1}, B_{2} is restricted to integers satisfying the conditions

B_{1}Δg_{1} ≤ Q, B_{2}Δg_{2} ≤ Q, G_{1} < g_{1} ≤ 2G_{1}, G_{2} < g_{2} ≤ 2G_{2},

|v|(pp′)^{−1}P^{2−ε} < B_{1}B_{2}Δg_{1}g_{2}, ρ_{d}(ΔB_{1}) = ρ_{d}(ΔB_{2}) = ρ_{d}(Δ), (213)

(g_{1}g_{2}, 2pp′d_{1}d_{2}d_{3}ΔB_{1}B_{2}) = (g_{1}, g_{2}) = (B_{1}B_{2}Δ, pp′) = (B_{1}, B_{2}) = 1.

Here we have to verify that if ρ_{d}(ΔB_{1}) = ρ_{d}(ΔB_{2}) then the common value of these integers equals ρ_{d}(Δ). This follows easily from the definition of ρ_{d} and from the properties of the integers involved.

We shall estimate E(G_{1}, G_{2}) by two different ways.

The first estimate for E(G_{1}, G_{2}). Using Lemma 2 (ii) and (iii) we find that S_{d}(ΔB_{j}, −β_{j}) ≪ (ΔB_{j})^{3/2} d_{1}d_{2}d_{3} (214) and it is easy to see that if (μ, ΔB_{1}B_{2}) = 1 then

∑_{\begin{matrix}β_{1}(ΔB_{1})^{∗} , β_{2}(ΔB_{2})^{∗} \\ μ(β_{1}B_{2}+β_{2}B_{1})≡v (ΔB_{1}B_{2}) \end{matrix}} 1 ≪ Δ. (215)

From (211), (214) and (215) we get Ξ(μ) ≪ Δ^{4}(B_{1}B_{2})^{3/2} (d_{1}d_{2}d_{3})^{2} if (μ, ΔB_{1}B_{2}) = 1 . (216) Applying Lemma 2 (iii) we get

∑_{G_{i}<g_{i}≤2G_{i}} g_{i}^{−3}|S^{3}(g_{i}, 1)| ≪ ∑_{G_{i}<g_{i}≤2G_{i}} g_{i}^{−3/2} ≪ G_{i}^{−1/2} , i = 1, 2 . (217)

From (209), (212), (216) and (217) we obtain E(G_{1}, G_{2}) ≪ E^{#}(G_{1}, G_{2}) , (218) where

E^{#}(G_{1}, G_{2}) = P^{4}R(G_{1}G_{2})^{−1/2} ∑_{(D)} d_{1}d_{2}d_{3} T , (219)

and

T = ∑_{\begin{matrix}Δv≤4R^{2}Q^{2}P^{ε−2} \\ Δ,v>0 \end{matrix}} Δ^{−2} ∑_{B_{1},B_{2} (221)}(B_{1}B_{2})^{−3/2} . (220)

Here we sum over integers B_{1} and B_{2} satisfying

B_{1} ≤ QΔ^{−1}G_{1}^{−1}, B_{2} ≤ QΔ^{−1}G_{2}^{−1},

|v| (16R^{2}ΔG_{1}G_{2})^{−1}P^{2−ε} ≤ B_{1}B_{2}, (B_{1}B_{2}, pp′) = 1, (221)

ρ_{d}(ΔB_{1}) = ρ_{d}(ΔB_{2}) = ρ_{d}(Δ) .

Using these conditions we get

(222)  T ≪ ∑_{\begin{matrix}Δv≤4R^{2}Q^{2}P^{ε−2} \\ Δ,v>0 \end{matrix}} Δ^{−2}(vR^{−2}(ΔG_{1}G_{2})^{−1}P^{2−ε})^{−3/2} T_{1}^{2}

≪ R^{3} (G_{1}G_{2})^{3/2} P^{2ε−3} ∑_{0<Δ≤4R^{2}Q^{2}P^{ε−2}} Δ^{−1/2} T_{1}^{2} ,

where

T_{1} = ∑_{\begin{matrix}B≤P \\ ρ_{d}(ΔB)=ρ_{d}(Δ) \end{matrix}} 1 . (223)

Consider T_{1}. If ρ_{d}(ΔB) = ρ_{d}(Δ) then every prime factor of B divides also 2d_{1}d_{2}d_{3}Δ. Hence

(224) T_{1} ≤ ∑_{\begin{matrix}B=1 \\ p|B⇒p|2d_{1}d_{2}d_{3}Δ \end{matrix}}^{∞}(P B^{−1})^{ε} = P^{ε} ∏_{p|2d_{1}d_{2}d_{3}Δ} (1 + p^{−ε} + p^{−2ε} + …)
≤ P^{ε} (1 + 2^{−ε} + 2^{−2ε} + …)^{ν(2d_{1}d_{2}d_{3}Δ)} ≪ P^{2ε} .

We substitute this estimate for T_{1} in (222) and then use (219) to get E^{#}(G_{1}, G_{2}) ≪ G_{1}G_{2} R^{5} Q D^{6} P^{10ε} . (225) From this inequality and (218) we obtain E(G_{1}, G_{2}) ≪ G_{1}G_{2} R^{5} Q D^{6} P^{10ε} . (226) The second estimate for E(G_{1}, G_{2}). Suppose that G_{1} ≤ G_{2}. (227) We use (209), (211), (212), and Lemma 2 (iii) to get

(228) E(G_{1}, G_{2}) ≪ P^{2}R^{−1} ∑_{(R)} ∑_{(D)}(d_{1}d_{2}d_{3})^{−1} ∑_{0<n<N} ∑_{\begin{matrix}0<Δ|v|≤4R^{2}Q^{2}P^{ε−2} \\ (Δv,pp′)=1 \end{matrix}}

× ∑_{B_{1},B_{2} (221)}(B_{1}B_{2}Δ^{2})^{−3}

× ∑_{\begin{matrix}G_{1}<g_{1}≤2G_{1} \\ (g_{1},2pp′d_{1}d_{2}d_{3}ΔB_{1}B_{2})=1 \end{matrix}} g_{1}^{−3/2} max_{\begin{matrix}G,G′: \\ G_{2}≤G≤G′≤2G_{2} \end{matrix}} |M_{0}|,

where

M_{0} = ∑_{\begin{matrix}G<g≤G′ \\ (g,2pp′d_{1}d_{2}d_{3}ΔB_{1}B_{2}g_{1})=1 \end{matrix}} g^{−3} S^{3}(g, 1) · e(\frac{nv}{B_{1}B_{2}Δg_{1}gpp′} )

× (\frac{g}{pp′} ) · e(\frac{−Nv\overline{(B_{1}B_{2}Δg_{1}g)}}{pp′}) · (\frac{g}{g_{1}} ) · (\frac{−pp′B_{1}B_{2}g_{1}Δv}{g} ) · Ξ(pp′g_{1}g) .

We have

M_{0} = ∑_{λ(ΔB_{1}B_{2})^{∗}} Ξ(pp′g_{1}λ) M_{1} , (229)

M_{1} = ∑_{\begin{matrix}G<g≤G′ \\ (g,2pp′d_{1}d_{2}d_{3}g_{1})=1 \\ g≡λ (ΔB_{1}B_{2}) \end{matrix}} g^{−3} S^{3}(g, 1) e(\frac{nv}{B_{1}B_{2}Δg_{1}gpp′} ) · (\frac{g}{pp′} )

× e(\frac{−Nv\overline{(B_{1}B_{2}Δg_{1}g)}}{pp′}) · (\frac{g}{g_{1}} ) · (\frac{−pp′B_{1}B_{2}g_{1}Δv}{g} ).

From (216) and (229) we get

M_{0} ≪ Δ^{4} (B_{1}B_{2})^{3/2} (d_{1}d_{2}d_{3})^{2} ∑_{λ (ΔB_{1}B_{2})^{∗}} |M_{1}|. (230)

We divide M_{1} into four parts according to the congruence class of g modulo 8. We have

M_{1} ≪ max_{j=1,3,5,7} |M_{2}(j)| , (231)

where the summation in M_{2}(j) is restricted to g ≡ j (8).

Define the natural numbers B′_{1}, B′_{2}, Δ′, v′ by (B′_{1}B′_{2}Δ′v′, 2) = 1 and B_{1} = 2^{μ_{1}}B′_{1}, B_{2} = 2^{μ_{2}}B′_{2}, Δ = 2^{μ_{3}}Δ′, |v| = 2^{μ_{4}}v′. Consider any of the sums M_{2}(j). We use Lemmas 2 (iv) and 7 to get

|M_{2}(j)| = |M_{3}|, (232)

where

M_{3} = ∑_{\begin{matrix}G<g≤G′ \\ (g,pp′d_{1}d_{2}d_{3}g_{1})=1 \\ g≡λ (ΔB_{1}B_{2}) \\ g≡j (8) \end{matrix}} g^{−3/2}e(\frac{nv}{B_{1}B_{2}Δg_{1}gpp′} )

× e(\frac{−Nv\overline{(B_{1}B_{2}Δg_{1}g)}}{pp′}) (\frac{B′_{1}B′_{2}Δ′v′}{g} ) .

We may assume that j ≡ λ ((8, B_{1}B_{2}Δ)) because otherwise the sum M_{3} would be empty. Denote

T = [8, B_{1}B_{2}Δ] . (233)

There exists π satisfying (π, T) = 1 and such that the system of congruences g ≡ λ (ΔB_{1}B_{2}) , g ≡ j (8) is equivalent to g ≡ π (T). We apply again Lemma 7 to get

|M_{3}| = |M_{4}|, (234)

where

M_{4} = ∑_{\begin{matrix}G<g≤G′ \\ (g,pp′d_{1}d_{2}d_{3}g_{1})=1 \\ g≡π (T) \end{matrix}} g^{−3/2}e(\frac{nv}{B_{1}B_{2}Δg_{1}gpp′} )e(\frac{−Nv\overline{(B_{1}B_{2}Δg_{1}g)}}{pp′})(\frac{g}{v′} ).

Furthermore, we use (221) and easily find that

\frac{d}{dx}(x^{−3/2} e(\frac{nv}{B_{1}B_{2}Δg_{1}xpp′} ) ) ≪ P^{ε}G_{2}^{−5/2}

uniformly for x ∈ [G_{2}, 2G_{2}]. We apply Abel’s formula to get

M_{4} ≪ P^{ε}G_{2}^{−3/2} max_{G′′∈[G,G′]} |M_{5}|, (235)

where

M_{5} = ∑_{\begin{matrix}G<g≤G′′ \\ (g,pp′d_{1}d_{2}d_{3}g_{1})=1 \\ g≡π (T) \end{matrix}} e(\frac{−Nv\overline{(B_{1}B_{2}Δg_{1}g)}}{pp′})(\frac{g}{v′} ).

Finally, to get rid of the condition (g, d_{1}d_{2}d_{3}g_{1}) = 1, imposed on g, we use the well known identity for the Möbius function and obtain

(236)  M_{5} = ∑_{\begin{matrix}G<g≤G′′ \\ (g,pp′)=1 \\ g≡π (T) \end{matrix}} ( ∑_{δ|(g,d_{1}d_{2}d_{3}g_{1})} μ(δ))e(\frac{−Nv\overline{(B_{1}B_{2}Δg_{1}g)}}{pp′})(\frac{g}{v′} )

= ∑_{δ|d_{1}d_{2}d_{3}g_{1}} μ(δ) M_{6} ,

where

M_{6} = ∑_{\begin{matrix}G<g≤G′′ \\ (g,pp′)=1 \\ g≡π (T) \\ g≡0 (δ) \end{matrix}} e(\frac{−Nv\overline{(B_{1}B_{2}Δg_{1}g)}}{pp′})(\frac{g}{v′} ) .

We have (π, T) = 1. Hence we can assume that (δ, T) = 1 because otherwise the sum M_{6} would be empty. Therefore we obtain

|M_{6}| ≤ |M_{7}| , (237)

where

M_{7} = ∑_{\begin{matrix}Gδ^{−1}<g≤G′′δ^{−1} \\ (g,pp′)=1 \\ g≡π′ (T) \end{matrix}} e(\frac{M\overline{g}}{pp′} ) (\frac{g}{v′} )

and

M ≡ −Nv \overline{(B_{1}B_{2}Δg_{1}δ)} (pp′) , π′ ≡ π \overline{δ} (T) .

To estimate M_{7} we divide it into O(G_{2}(δv′T pp′)^{−1}) complete sums and at most one incomplete sum modulo v′T pp′. We get

M_{7} ≪ G_{2} (δv′T pp′)^{−1} |M_{8}| + max_{\begin{matrix}H, H′ \\ 0<H′−H<v′T pp′ \end{matrix}} |M_{9}| , (238)

where

M_{8} = ∑_{\begin{matrix}g(v′T pp′) \\ (g,pp′)=1 \\ g≡π′ (T) \end{matrix}} e(\frac{M\overline{g}}{pp′} )(\frac{g}{v′} ) , M_{9} = ∑_{\begin{matrix}H<g≤H′ \\ (g,pp′)=1 \\ g≡π′ (T) \end{matrix}} e(\frac{M\overline{g}}{pp′} )(\frac{g}{v′} ) .

Consider the complete sum M_{8}. First we note that the relevant values of g are all coprime to v′T pp′. Obviously (v′T, pp′) = 1. Hence we have

M_{8} = ∑_{\begin{matrix}h_{1}(v′T)^{∗} , h_{2}(pp′)^{∗} \\ h_{1}pp′+h_{2}v′T ≡π′ (T) \end{matrix}} e(\frac{M\overline{(h_{1}pp′ + h_{2}v′T)}}{pp′}) (\frac{h_{1}pp′ + h_{2}v′T}{v′} )

= ∑_{\begin{matrix}h_{1}(v′T)^{∗} \\ h_{1}pp′≡π′ (T) \end{matrix}} (\frac{h_{1}pp′}{v′} ) c_{pp′}(M \overline{v′T}) ,

where c_{pp′} is the Ramanujan sum, defined by (4). To estimate it we apply Lemma 5 . We also note that the sum over h_{1} has O(v′) terms. Therefore we obtain

M_{8} ≪ v′ . (239)

Consider now the incomplete sum M_{9}. We treat it in the following standard way:

M_{9} = ∑_{\begin{matrix}g(v′T pp′) \\ (g,pp′)=1 \\ g≡π′ (T) \end{matrix}} e(\frac{M\overline{g}}{pp′} ) (\frac{g}{v′} ) ∑_{H<s≤H′}(v′T pp′)^{−1} ∑_{h(v′T pp′)} e(\frac{h(g − s)}{v′T pp′})

= (v′T pp′)^{−1} ∑_{h(v′T pp′)} ( ∑_{H<s≤H′} e(\frac{−hs}{v′T pp′} ) ) M_{10}(h) ,

where

M_{10}(h) = ∑_{\begin{matrix}g(v′T pp′) \\ (g,pp′)=1 \\ g≡π′ (T) \end{matrix}} (\frac{g}{v′} )e(\frac{M\overline{g}}{pp′} + \frac{hg}{v′T pp′} ) .

It is clear that

M_{9} ≪ ∑_{|h|≤v′T pp′}(1 + |h|)^{−1} |M_{10}(h)| . (240)

It remains to estimate M_{10}(h). Again we note that the relevant values of g are all coprime to v′T pp′ and that (v′T, pp′) = 1. Hence we have

M_{10}(h) = ∑_{\begin{matrix}t_{1}(v′T)^{∗} , t_{2}(pp′) \\ t_{1}pp′+t_{2}v′T ≡π′ (T) \end{matrix}} (\frac{t_{1}pp′ + t_{2}v′T}{v′} ) e(\frac{M\overline{(v′T t_{2})}}{pp′} + \frac{t_{1}h}{v′T} + \frac{t_{2}h}{pp′})

= ∑_{\begin{matrix}t_{1}(v′T)^{∗} \\ t_{1}pp′≡π′ (T) \end{matrix}} (\frac{t_{1}pp′}{v′} ) e(\frac{t_{1}h}{v′T} ) K(pp′, h, M \overline{v′T}) ,

where K is the Kloosterman sum, defined by (4). We estimate it using Lemma 4. Obviously, the sum over t_{1} has O(v′) terms. Hence M_{10}(h) ≪ v′R . (241) Inequalities (240) and (241) imply M_{9} ≪ v′RP^{ε} . (242) From (238), (239) and (242) we get M_{7} ≪ (G_{2}(δT R^{2})^{−1} + v′R)P^{ε} . (243)

We take into account (231) – (237) and (243) to find that M_{1} ≪ ( (G_{2}^{1/2}ΔB_{1}B_{2}R^{2})^{−1} + |v| R G_{2}^{−3/2} )P^{3ε} . (244) To estimate E(G_{1}, G_{2}) we use (228), (230) and (244). We get E(G_{1}, G_{2}) ≪ E^{(1)} + E^{(2)}, (245) where E^{(j)}, j = 1, 2, are the contributions to E(G_{1}, G_{2}) coming from the first, respectively, the second summand from the right hand side of (244). We use (219), (228), (230) and (244) to find that E^{(1)} ≪ P^{3ε}R^{−2}E^{#}(G_{1}, G_{2}) . (246) Now we apply (225) to get E^{(1)} ≪ G_{1}G_{2}R^{3}D^{6}QP^{15ε}. (247)

Consider E^{(2)}. We use (228), (230) and (244) to get

E^{(2)} ≪ P^{4+3ε} R^{2} G_{1}^{−1/2} G_{2}^{−3/2} ∑_{(D)} d_{1}d_{2}d_{3} T^{∗} , (248)

where

T^{∗} = ∑_{\begin{matrix}Δv≤4R^{2}Q^{2}P^{ε−2} \\ Δ,v>0 \end{matrix}} Δ^{−1}v ∑_{B_{1},B_{2} (221)}(B_{1}B_{2})^{−1/2}.

It is clear by analogy with (222) that

T^{∗} ≪ ∑_{\begin{matrix}Δv≤4R^{2}Q^{2}P^{ε−2} \\ Δ,v>0 \end{matrix}} Δ^{−1}v (vR^{−2}(ΔG_{1}G_{2})^{−1}P^{2−ε})^{−1/2} T_{1}^{2} , (249)

where T_{1} is defined by (223). Using (224), (248) and (249) we easily find that

E^{(2)} ≪ R^{6} Q^{3} G_{2}^{−1} D^{6} P^{10ε} . (250)

From (227), (245), (247) and (250) we find

E(G_{1}, G_{2}) ≪ (G_{1}G_{2}R^{3}Q + R^{6}Q^{3}( max(G_{1}, G_{2}))^{−1}) D^{6}P^{15ε}. (251)

Conclusion. Let H be a parameter which we choose below. If G_{1}, G_{2} ≤ H then we use the first estimate (226) for E(G_{1}, G_{2}). If H < max(G_{1}, G_{2}) ≤ Q then we use the second estimate, given by (251). We obtain

E(G_{1}, G_{2}) ≪ (R^{5}H^{2}Q + R^{3}Q^{3} + R^{6}H^{−1}Q^{3}) D^{6}P^{15ε}. (252)

We choose H = R^{1/3}Q^{2/3} and use (208) and (252) to obtain

E^{∗} ≪ (R^{3}Q^{3} + R^{17/3}Q^{7/3}) D^{6}P^{16ε}. (253)

3.4.7 The estimate for E_{1}.

Let us recall the main results of Section 3.4. Equation (84) states that

E_{1} = E_{1}^{(1)} − 2E_{1}^{(2)} + E_{1}^{(3)} .

The quantities, involved in the right hand side of this formula, satisfy the asymptotic formulas (109), (111) and (121), respectively:

E_{1}^{(1)} = U_{1} + O(V^{∗}) ,
E_{1}^{(2)} = U_{1} + O(V^{∗}) + O(D^{∗}) ,
E_{1}^{(3)} = U_{1} + O(D^{∗}) + O(E^{∗}) .

The quantities, included in the O – terms, satisfy (193), (194) and (253), respectively:

V^{∗} ≪ (P^{3}R + P^{2}R^{5})D^{6}P^{15ε} ,
D^{∗} ≪ P^{4+5ε}Q^{−1}D^{3}R ,
E^{∗} ≪ (R^{3}Q^{3} + R^{17/3}Q^{7/3}) D^{6}P^{16ε} .

Therefore we obtain

(254)  E_{1} ≪ (P^{3}D^{6}R + P^{2}D^{6}R^{5}
+ P^{4}Q^{−1}D^{3}R + Q^{3}D^{6}R^{3} + Q^{7/3}D^{6}R^{17/3})P^{20ε} .

3.5 Proof of Proposition 1.

Consider the sum E(D, Q), defined by (31). From (41) and (43) we get

E(D, Q) ≪ (P R^{−2}|E_{1}| + P R^{−1}E_{2})^{1/2}P^{ε} .

To estimate E_{2} we use (82):

E_{2} ≪ (P^{3}D^{6} + P^{4}D^{3}Q^{−1})P^{9ε} .

Similarly, for E_{1} we use the estimate (254), given above. We obtain

(255)  E(D, Q) ≪ (P^{2}D^{3}R^{−1/2} + P^{3/2}D^{3}R^{3/2} + P^{5/2}Q^{−1/2}D^{3/2}R^{−1/2}
+ P^{1/2}Q^{3/2}D^{3}R^{1/2} + P^{1/2}Q^{7/6}D^{3}R^{11/6})P^{12ε} .

The parameters D, Q, R satisfy the conditions (39), (40), (42), (83) and (196):

Q ≤ P^{1−ε} ,
D = P^{α_{0}} where α_{0} ∈ (0, 1) ,
R = P^{α_{1}} where α_{1} ∈ (0, 1) ,
D ≤ R^{1/6}P^{−10ε} ,
P^{1+20ε}D^{3}R^{−1} ≤ Q ,
Q ≤ P^{1−ε}R^{−1/2} .

It is not difficult to see that the optimal choice (up to a power of P^{ε}) of these parameters is R = P^{5/23} , Q = P^{20/23} , D = P^{2/69 − 10ε} . (256) From (255) and (256) we obtain E(D, Q) ≪ P^{2−ε} . (257)

This completes the proof of Proposition 1.

□

3.6 Proof of Proposition 2.

3.6.1 Beginning. Let us write β(d) = β_{1}(d_{1})β_{2}(d_{2})β_{3}(d_{3}) for brevity and consider the sum

H_{1} = ∑_{(D)} β(d) L_{d}(N) . (258)

We use (28), (30), (34) and Proposition 1 to get

H_{1} = ∑_{(D)} β(d) ∑_{p≤P} Ω_{d}(N − p^{2}) = H_{2} + O(P^{2−ε}) , (259)

where

H_{2} = ∑_{(D)} β(d) ∑_{p≤P} M_{d,Q}(N − p^{2}) . (260)

Here M_{d,Q} is defined by (29) and Q satisfies (32). We use (29) and change the order of summation to find that

H_{2} = P ∑_{(D)} \frac{β(d)}{d_{1}d_{2}d_{3}} ∑_{q≤Q} q^{−3} ∑_{a(q)^{∗}} S_{d}(q, a)

× ∑_{t_{0}P <p≤P} H(1 − \frac{p^{2}}{N} ) e_{q}(a(p^{2} − N)) ,

where t_{0} = (1 − t_{2})^{1/2} ∈ (0, 1) and t_{2} is specified in Lemma 11. Now we apply Abel’s formula to get

H_{2} = −P ∫_{t_{0}P}^{P} B(x) \frac{d}{dx}( H(1 − \frac{x^{2}}{N} ) ) dx , (261)

where

B(x) = ∑_{(D)} \frac{β(d)}{d_{1}d_{2}d_{3}} ∑_{q≤Q} q^{−3} ∑_{a(q)^{∗}} S_{d}(q, a) Z(x) (262)

and

Z(x) = Z = ∑_{t_{0}P <p≤x} e_{q}(a(p^{2} − N)) . (263)

Consider the sum Z. We divide it into subsums according to the congruence class m of p modulo q. There is no contribution from m such that (m, q) > 1. Therefore we get

(264)  Z = ∑_{m(q)^{∗}} e_{q}(a(m^{2} − N))(π(x, q, m) − π(t_{0}P, q, m))

= \frac{1}{φ(q)} ∫_{t_{0}P}^{x} \frac{dt}{log t} e_{q}(−aN) T (q, a)

+ ∑_{m(q)^{∗}} e_{q}(a(m^{2} − N))(Δ(x, q, m) − Δ(t_{0}P, q, m))

where Δ(x, q, m) and T (q, a) are defined by (3) and (7), respectively.

From (262) – (264) we get

B(x) = B_{0} ∫_{t_{0}P}^{x} \frac{dt}{log t} + C(x) − C(t_{0}P) , (265)

where

(266)  C(x) = ∑_{(D)} \frac{β(d)}{d_{1}d_{2}d_{3}} ∑_{q≤Q} q^{−3}

× ∑_{a(q)^{∗}} S_{d}(q, a) ∑_{m(q)^{∗}} e_{q}(a(m^{2} − N))Δ(x, q, m) ,

(267)  B_{0} = ∑_{(D)} \frac{β(d)}{d_{1}d_{2}d_{3}} ∑_{q≤Q} h_{d}(q)

and where h_{d}(q) is defined by (9). 3.6.2 The estimation of C(x). Consider the sum C(x). We expect that it is negligible because it depends on the quantity Δ(x, q, m), which is small on average with respect to q and m. More precisely, we shall prove that if t_{0}P ≤ x ≤ P then we have C(x) ≪ P (log P)^{−A} . (268)

It is clear that

C(x) = ∑_{q≤Q} ∑_{m(q)^{∗}} Γ(q, m) Δ(x, q, m) ,

where

Γ(q, m) = q^{−3} ∑_{(D)} \frac{β(d)}{d_{1}d_{2}d_{3}} ζ_{d}(q, m) (269)

and

ζ_{d}(q, m) = ∑_{a(q)^{∗}} S_{d}(q, a) e_{q}(a(m^{2} − N)) . (270)

Obviously, Γ(q, m) is always real. Using Cauchy’s inequality we get |C(x)|^{2} ≤ L M , (271) where

(272)  L = ∑_{q≤Q} ∑_{m(q)^{∗}} Δ(x, q, m)^{2} ,
(273)  M = ∑_{q≤Q} ∑_{m(q)^{∗}} Γ(q, m)^{2} .

We estimate L using Lemma 1. Consider the sum M. We shall prove that

M ≪ (log P)^{C} (274)

for some absolute constant C > 0. Then we will take into account (271) and the proof of (268) will be complete.

From (269) and (270) we get

M = ∑_{\begin{matrix}d_{1},d_{2},d_{3}≤D \\ h_{1},h_{2},h_{3}≤D \end{matrix}} \frac{β(d) β(h)}{d_{1}d_{2}d_{3} h_{1}h_{2}h_{3}} ∑_{q≤Q} λ(q, d, h) , (275)

where

λ(q, d, h) = q^{−6} ∑_{m(q)^{∗}} ζ_{d}(q, m) ζ_{h}(q, m) . (276)

Applying Lemma 2 (i) we easily see that the function λ is multiplicative with respect to q.

From (35) we conclude that the relevant values of d and h are vectors with odd components. Furthermore, we use (270), (276) and Lemma 2 (vi) and find that

λ(2^{l}, d, h) ≪ 1 . (277)

Consider λ(p^{l}, d, h), where p > 2 is a prime. Assume that

p^{α_{i}} || d_{i} ,  p^{β_{i}} || h_{i} ,  ν_{i} = min(l, 2α_{i}) ,  μ_{i} = min(l, 2β_{i}) . (278)

Using (4) and Lemma 2 (ii) and (iv) we get

ζ_{d}(p^{l}, m) = ζ_{d}^{∗}(p^{l}, m) ∏_{i=1}^{3} ( p^{ν_{i}} S(p^{l−ν_{i}}, 1) ) , (279)

where

ζ_{d}^{∗}(p^{l}, m) = \begin{cases}c_{p^{l}}(m^{2} − N) & if ν_{1} + ν_{2} + ν_{3} ≡ l (2) , \\ ∑_{a(p^{l})^{∗}} (\frac{a}{p}) e_{p^{l}}(a(m^{2} − N)) & if ν_{1} + ν_{2} + ν_{3} ≢ l (2) . \end{cases} (280)

From (276), (278) – (280) and Lemma 2 (v) we obtain

(281)  |λ(p^{l}, d, h)| ≤ p^{−6l}( ∏_{i=1}^{3} p^{l+ \frac{1}{2} (ν_{i}+μ_{i})}) ∑_{m(p^{l})^{∗}} | ζ_{d}^{∗}(p^{l}, m) ζ_{h}^{∗}(p^{l}, m) |
≤ p^{−3l+ \frac{1}{2} (ν_{1}+ν_{2}+ν_{3}+μ_{1}+μ_{2}+μ_{3})}(T_{1} + T_{2}) ,

(282)  T_{1} = ∑_{m(p^{l})^{∗}} | c_{p^{l}}(m^{2} − N) |^{2} ,

(283)  T_{2} = ∑_{m(p^{l})^{∗}} | ∑_{a(p^{l})^{∗}} (\frac{a}{p})e_{p^{l}}(a(m^{2} − N)) |^{2} .

We shall prove that

T_{1} ≤ 3p^{2l} , T_{2} ≤ 3p^{2l} . (284)

These inequalities and (281) imply that for any prime p > 2 and for any integer l we have

|λ(p^{l}, d, h)| ≤ 6p^{−l+ \frac{1}{2} (ν_{1}+ν_{2}+ν_{3}+μ_{1}+μ_{2}+μ_{3})} . (285)

From (277), (278) and (285) we obtain

(286)  ∑_{q≤Q} λ(q, d, h) ≪ ∑_{2^{α}≤Q} ∑_{\begin{matrix}m≤Q \\ (m,2)=1 \end{matrix}} λ(m, d, h)

≪ (log P) ∑_{m≤Q} \frac{6^{ν(m)}}{m} ∏_{i=1}^{3}(m, d_{i})(m, h_{i}) .

Now we use (36), (275) and (286) to find that

M ≪ (log P) ∑_{q≤Q} \frac{6^{ν(q)}}{q} ( ∑_{d≤D} \frac{τ(d)(q, d)}{d})^{6} .

It is easy to see that the sum over d is ≪ τ^{2}(q) (log P)^{2}. Hence we get

M ≪ (log P)^{13} ∑_{q≤Q} \frac{6^{ν(q)}}{q} τ^{12}(q) ≪ (log P)^{13} ∑_{q≤Q} \frac{τ^{15}(q)}{q} ≪ (log P)^{2^{16}} ,

so we have established (274).

It remains to prove that if p > 2 is a prime and l is an integer then the inequalities (284) hold.

Consider T_{1}. Using the exact formula for the Ramanujan sum, given by Lemma 5, we get

T_{1} = ∑_{0≤h≤l} ∑_{\begin{matrix}m(p^{l})^{∗} \\ (p^{l},m^{2}−N)=p^{h} \end{matrix}} |\frac{φ(p^{l})}{φ(p^{l−h})}μ(p^{l−h})|^{2}

≤ p^{2l−2} ∑_{\begin{matrix}m(p^{l})^{∗} \\ m^{2}≡N (p^{l−1}) \end{matrix}} 1 + p^{2l}H_{N}(p^{l})

= p^{2l−1}H_{N}(p^{l−1}) + p^{2l}H_{N}(p^{l}) , where H_{N}(q) is the cardinality of the set { m(q)^{∗} | m^{2} ≡ N (q) }. It is well known that if p > 2 and p ∤ N then H_{N}(p^{l}) ≤ 2 for any l. Obviously, if p | N then H_{N}(p^{l}) = 0. The inequality (284) for T_{1} follows.

Consider T_{2}. Using Lemma 6 (ii) we easily obtain

∑_{a(p^{l})^{∗}} (\frac{a}{p}) e_{p^{l}}(an) = \begin{cases}p^{l−1} S(p, 1) (\frac{n/p^{l−1}}{p} ) & if p^{l−1} || n , \\ 0 & otherwise . \end{cases}

We apply Lemma 2 (v) and the estimate for H_{N}(p^{l}), given above, and find that

T_{2} ≤ ∑_{\begin{matrix}m(p^{l})^{∗} \\ (p^{l},m^{2}−N)=p^{l−1} \end{matrix}} |p^{l−1}S(p, 1)|^{2} ≤ p^{2l−1} ∑_{\begin{matrix}m(p^{l})^{∗} \\ (p^{l},m^{2}−N)=p^{l−1} \end{matrix}} 1

≤ p^{2l}H_{N}(p^{l−1}) ≤ 2p^{2l} .

So the inequalities (284) are established and the proof of (268) is complete.

3.6.3 The end of the proof of Proposition 2.

Let us consider the sum B_{0}, defined by (267). We extend the summation over q to infinity. Using (17) and (32) we find that

∑_{q>Q} |h_{d}(q)| ≪ (d_{1}d_{2}d_{3})^{3} ∑_{q>Q} \frac{(q, N)}{q^{2−ε}} ≪ D^{9} Q^{−1} P^{2ε} ≪ P^{−1/5} . (287)

Hence

B_{0} = B_{1} + O(P^{−ε}) , (288)

where

B_{1} = ∑_{(D)} \frac{β(d)}{d_{1}d_{2}d_{3}} Σ_{0} (289)

and where Σ_{0} is defined by (16). From (265), (268) and (288) we find that if t_{0}P ≤ x ≤ P then

B(x) = B_{1} ∫_{t_{0}P}^{x} \frac{dt}{log t} + O(P (log P)^{−A}) . (290)

We substitute this expression for B(x) in (261). The contribution to H_{2} coming from the remainder term is

≪ P^{2}(log P)^{−A} ∫_{t_{0}P}^{P} |H′(1 − \frac{x^{2}}{P^{2}} )\frac{2x}{P^{2}} |dx ≪ P^{2}(log P)^{−A} .

We integrate by parts and find that the contribution to H_{2} from the main term of (290) is N_{0} B_{1}, where N_{0} is defined by (26). Now we apply (259) to find that

H_{1} = N_{0}B_{1} + O(P^{2}(log P)^{−A}) . (291)

We take into account (37), (258), (289) and (291) and the proof of Proposition 2 is complete.

□

4 The proof of Theorem 1.

Consider the sum

𝔉 = ∑_{\begin{matrix}p^{2}+x_{1}^{2}+x_{2}^{2}+x_{3}^{2}=N \\ (x_{i},𝔓)=1 \end{matrix}} ω(x) ,

where

𝔓 = ∏_{2<p<z_{1}} p  and  z_{1} = P^{α} (292)

for some α ∈ (0, 1), which we shall specify later. Using the condition N ≡ 4 (24) and the definition of ω(x) we find that the solutions of (2), such that 2 | px_{1}x_{2}x_{3}, are not counted in 𝔉.

Suppose that

𝔉 ≫ P^{2}(log P)^{−4} . (293)

Then there exist constants c_{0} > 0 and N_{0} > 0 such that for any integer N > N_{0} there are at least c_{0}P^{2}(log P)^{−4} quadruples p, x_{1}, x_{2}, x_{3} satisfying (2) and such that p is a prime and each x_{i} is an almost–prime with no more than α^{−1} prime factors, counted according to multiplicity.

So, our aim is to establish (293) with α as large as possible. We apply the vector sieve proposed by Iwaniec [11] and used also by Brüdern and Fouvry [1], [2] as well as by the second author [18], [19]. In many places we omit the calculations because they are similar to those in [1], [18] and [19].

It is convenient to sieve by the small primes separately. Define

z_{0} = (log P)^{1000} ,  𝔓_{0} = ∏_{2<p<z_{0}} p ,  𝔓_{1} = ∏_{z_{0}≤p<z_{1}} p . (294)

We represent the sum 𝔉 as

𝔉 = ∑_{p^{2}+x_{1}^{2}+x_{2}^{2}+x_{3}^{2}=N} ω(x) Φ_{1} Φ_{2} Φ_{3} Λ_{1} Λ_{2} Λ_{3} , (295)

Φ_{i} = ∑_{k|(x_{i},𝔓_{0})} μ(k) , Λ_{i} = ∑_{l|(x_{i},𝔓_{1})} μ(l) .

Let

D_{0} = P^{ε} ,  D_{1} = P^{2/69−11ε} ,  D = D_{0}D_{1} . (296)

We define

(297)  s_{0} = \frac{log D_{0}}{log z_{0}} = \frac{ε log P}{1000 log log P} ,
(298)  s_{1} = \frac{log D_{1}}{log z_{1}} = (\frac{2}{69} − 11ε) α^{−1} .

To apply the sieve method we need the inequalities s_{0} > 2 and s_{1} > 2. The first of them is obvious. To have the second we assume that α < 1/69 and take ε suffisiently small.

Consider Rosser’s weights λ_{i}^{±}(d) of orders D_{i} , i = 0, 1 . Define

Φ_{i}^{±} = ∑_{k|(x_{i},𝔓_{0})} λ_{0}^{±}(k) ,  Λ_{i}^{±} = ∑_{l|(x_{i},𝔓_{1})} λ_{1}^{±}(l) ,  i = 1, 2, 3 . (299)

The definition and the properties of the Rosser weights can be found in Iwaniec [12], [13]. In particular, we have

|λ_{i}^{±}(d)| ≤ 1 ; λ_{i}^{±}(d) = 0 if μ(d) = 0 or d > D_{i}; (300)

Φ_{i}^{−} ≤ Φ_{i} ≤ Φ_{i}^{+} ,  Λ_{i}^{−} ≤ Λ_{i} ≤ Λ_{i}^{+} ,  i = 1, 2, 3 . (301)

From (301) we easily get

(302) Φ_{1}Φ_{2}Φ_{3}Λ_{1}Λ_{2}Λ_{3} ≥ Φ_{1}^{−}Φ_{2}^{+}Φ_{3}^{+}Λ_{1}^{+}Λ_{2}^{+}Λ_{3}^{+} + Φ_{1}^{+}Φ_{2}^{−}Φ_{3}^{+}Λ_{1}^{+}Λ_{2}^{+}Λ_{3}^{+}

+ Φ_{1}^{+}Φ_{2}^{+}Φ_{3}^{−}Λ_{1}^{+}Λ_{2}^{+}Λ_{3}^{+} + Φ_{1}^{+}Φ_{2}^{+}Φ_{3}^{+}Λ_{1}^{−}Λ_{2}^{+}Λ_{3}^{+} + Φ_{1}^{+}Φ_{2}^{+}Φ_{3}^{+}Λ_{1}^{+}Λ_{2}^{−}Λ_{3}^{+}

+ Φ_{1}^{+}Φ_{2}^{+}Φ_{3}^{+}Λ_{1}^{+}Λ_{2}^{+}Λ_{3}^{−} − 5Φ_{1}^{+}Φ_{2}^{+}Φ_{3}^{+}Λ_{1}^{+}Λ_{2}^{+}Λ_{3}^{+} .

The proof of (302) is elementary and similar to the proof of Lemma 13 of [1]. From (295) and (302) we find that

𝔉 ≥ ∑_{i=1}^{6} 𝔉_{i} − 5𝔉_{7} , (303)

where 𝔉_{i} , 1 ≤ i ≤ 7 are the contributions of the consecutive terms from the right hand side of (302). Obviously 𝔉_{1} = 𝔉_{2} = 𝔉_{3} and 𝔉_{4} = 𝔉_{5} = 𝔉_{6}.

Consider, for example, 𝔉_{1}. We use (296), (299), (300) and change the order of summation to get

𝔉_{1} = ∑_{d_{1},d_{2},d_{3}≤D} β_{1}(d_{1})β_{2}(d_{2})β_{3}(d_{3}) L_{d}(N) ,

where L_{d}(N) is defined by (34) and

(304)  β_{1}(d_{1}) = ∑_{\begin{matrix}k|𝔓_{0} , l|𝔓_{1} \\ kl=d_{1} \end{matrix}} λ_{0}^{−}(k) λ_{1}^{+}(l) ,
(305)  β_{i}(d_{i}) = ∑_{\begin{matrix}k|𝔓_{0} , l|𝔓_{1} \\ kl=d_{i} \end{matrix}} λ_{0}^{+}(k) λ_{1}^{+}(l) , i = 2, 3 .

It is obvious that the functions β_{i} satisfy (35) and (36). Therefore, we can apply Proposition 2 to get 𝔉_{1} = 𝔉_{1}^{∗} + O(P^{2}(log P)^{−A}) , (306) where

𝔉_{1}^{∗} = ∑_{d_{1},d_{2},d_{3}≤D} \frac{β_{1}(d_{1})β_{2}(d_{2})β_{3}(d_{3})}{d_{1}d_{2}d_{3}} N_{0}(N) Σ_{0}(d, N) (307)

and where Σ_{0} and N_{0} are defined by (16) and (26).

Suppose that d_{i} are squarefree odd numbers and consider Σ_{0}. Using the identity (16) we find that Σ_{0}(d, N) = ξ_{0}(N) ℜ(d, N) , (308) where

(309)  ξ_{0} = ξ_{0}(N) = ∏_{p>2}(1 + h_{0}(p)) ,
(310) ℜ(d, N) = ∏_{p||d_{1}d_{2}d_{3}} \frac{1 + h_{1}(p)}{1 + h_{0}(p)} ∏_{p^{2}||d_{1}d_{2}d_{3}} \frac{1 + h_{2}(p)}{1 + h_{0}(p)} ∏_{p^{3}||d_{1}d_{2}d_{3}} \frac{1 + h_{3}(p)}{1 + h_{0}(p)} .

From (10) – (13), (309) and (310) we get 1 ≪ ξ_{0} ≪ log log P (311) and ℜ(d, N) ≪ τ^{2}(d_{1}) τ^{2}(d_{2}) τ^{2}(d_{3}) . (312) The calculations are standard and we leave them to the reader.

Suppose that d_{i} = k_{i}l_{i}, where k_{i} | 𝔓_{0}, l_{i} | 𝔓_{1}, i = 1, 2, 3, and denote by k and l the triples k_{1}, k_{2}, k_{3} and l_{1}, l_{2}, l_{3}, respectively. It is clear that ℜ(d, N) = ℜ(k, N) ℜ(l, N) . (313) Define

(314)  H^{±} = ∑_{k_{1},k_{2},k_{3}|𝔓_{0}} \frac{λ_{0}^{±}(k_{1})λ_{0}^{+}(k_{2})λ_{0}^{+}(k_{3})}{k_{1}k_{2}k_{3}} ℜ(k, N)
(315)  G^{±} = ∑_{l_{1},l_{2},l_{3}|𝔓_{1}} \frac{λ_{1}^{±}(l_{1})λ_{1}^{+}(l_{2})λ_{1}^{+}(l_{3})}{l_{1}l_{2}l_{3}} ℜ(l, N) .

From (304), (305), (307), (308) and (313) – (315) we find that 𝔉_{1}^{∗} = N_{0} ξ_{0} H^{−} G^{+} . (316) We study the other sums 𝔉_{i} in the same manner and use the analogs of (306) and (316) to obtain

𝔉_{1} = 𝔉_{2} = 𝔉_{3} = N_{0} ξ_{0} H^{−} G^{+} + O(P^{2}(log P)^{−A}) ,
𝔉_{4} = 𝔉_{5} = 𝔉_{6} = N_{0} ξ_{0} H^{+} G^{−} + O(P^{2}(log P)^{−A}) ,
𝔉_{7} = N_{0} ξ_{0} H^{+} G^{+} + O(P^{2}(log P)^{−A}) .

Now we apply (303) to get 𝔉 ≥ N_{0} ξ_{0} (3H^{−}G^{+} + 3H^{+}G^{−} − 5H^{+}G^{+}) + O(P^{2}(log P)^{−A}) . (317)

From (297) we see that s_{0} → ∞ as N → ∞. Hence the Rosser weights λ_{0}^{±} behave like the Möbius function and we can expect that the sums H^{±} can be approximated by the sum

H_{0} = ∑_{k_{1},k_{2},k_{3}|𝔓_{0}} \frac{μ(k_{1})μ(k_{2})μ(k_{3})}{k_{1}k_{2}k_{3}} ℜ(k, N) .

More precisely, the following asymptotic formula holds: H^{±} = H_{0} + O(exp(−\sqrt{log P})) . (318) We omit the proof because it does not differ significantly from the proof of Lemma 14 of [18] or formula (3.17) of [19].

It is easy to see that

H_{0} = ∏_{2<p<z_{0}} (1 − \frac{3(1 + h_{1}(p))}{p(1 + h_{0}(p))} + \frac{3(1 + h_{2}(p))}{p^{2}(1 + h_{0}(p))} − \frac{1 + h_{3}(p)}{p^{3}(1 + h_{0}(p))}) . (319)

From this formula, (10) – (13) and (294) we find that H_{0} ≍ (log log P)^{−3} . (320) We leave the easy verification of (319) and (320) to the reader.

Using (292), (294), (300), (312) and (315) we get

G^{±} ≪ ( ∑_{l|𝔓_{1}} \frac{τ^{2}(l)}{l})^{3} ≪ (log P)^{12} . (321)

From (27), (311), (317), (318) and (321) we find that 𝔉 ≥ N_{0} ξ_{0} H_{0} (3G^{−} − 2G^{+}) + O(P^{2}(log P)^{−A}) . (322)

It remains to estimate from below the difference 3G^{−} − 2G^{+}. Using (315) we get

(323) 3G^{−} − 2G^{+} = ∑_{l_{1},l_{2},l_{3}|𝔓_{1}} \frac{(3λ_{1}^{−}(l_{1}) − 2λ_{1}^{+}(l_{1})) λ_{1}^{+}(l_{2}) λ_{1}^{+}(l_{3})}{l_{1}l_{2}l_{3}} ℜ(l, N)
= W_{1} + W′_{1} ,

where in W_{1} we sum over l_{1}, l_{2}, l_{3} | 𝔓_{1} such that

(l_{1}, l_{2}) = (l_{1}, l_{3}) = (l_{2}, l_{3}) = 1 (324)

and where W′_{1} is the contribution of the other terms. We use the definition of 𝔓_{1}, given by (294), and find that if l_{i}, l_{j} | 𝔓_{1} and (l_{i}, l_{j}) > 1 then (l_{i}, l_{j}) ≥ z_{0}. From this observation, (300) and (312) we easily get

W′_{1} ≪ (log P)^{100} z_{0}^{−1} . (325)

Consider W_{1}. If the condition (324) holds then we have

ℜ(l, N) = ψ(l_{1})ψ(l_{2})ψ(l_{3}) ,

where

ψ(l) = ∏_{\begin{matrix}p|l \\ p>2 \end{matrix}} \frac{1 + h_{1}(p)}{1 + h_{0}(p)} . (326)

Hence

(327) W_{1} = ∑_{l_{1},l_{2},l_{3}|𝔓_{1}} \frac{(3λ_{1}^{−}(l_{1}) − 2λ_{1}^{+}(l_{1})) λ_{1}^{+}(l_{2}) λ_{1}^{+}(l_{3})}{l_{1}l_{2}l_{3}} ψ(l_{1})ψ(l_{2})ψ(l_{3})

× ∑_{\begin{matrix}h_{1}|(l_{2},l_{3}) \\ h_{2}|(l_{1},l_{3}) \\ h_{3}|(l_{1},l_{2}) \end{matrix}} μ(h_{1})μ(h_{2})μ(h_{3})

= ∑_{h_{1},h_{2},h_{3}|𝔓_{1}} μ(h_{1})μ(h_{2})μ(h_{3})

× ∑_{\begin{matrix}l_{1},l_{2},l_{3}|𝔓_{1} \\ l_{1}≡0([h_{2},h_{3}]) \\ l_{2}≡0([h_{1},h_{3}]) \\ l_{3}≡0([h_{1},h_{2}]) \end{matrix}} \frac{(3λ_{1}^{−}(l_{1}) − 2λ_{1}^{+}(l_{1})) λ_{1}^{+}(l_{2}) λ_{1}^{+}(l_{3})}{l_{1}l_{2}l_{3}} ψ(l_{1})ψ(l_{2})ψ(l_{3})

= W_{2} + W′_{2} ,

where W_{2} is the contribution of the terms with h_{1} = h_{2} = h_{3} = 1 and W′_{2} comes from the other terms.

Consider W′_{2}. If h_{i} | 𝔓_{1} and h_{i} > 1 then h_{i} ≥ z_{0}. Therefore, after some calculations, which we leave to the reader, we obtain W′_{2} ≪ (log P)^{100} z_{0}^{−1} . (328)

Consider W_{2}. Obviously W_{2} = (3𝔗^{−} − 2𝔗^{+}) (𝔗^{+})^{2} , (329) where

𝔗^{±} = ∑_{l|𝔓_{1}} \frac{λ_{1}^{±}(l)ψ(l)}{l} .

Using Lemma 10 of [1] we establish the inequalities

(330)  𝔑 ≤ 𝔗^{+} ≤ 𝔑 {F (s_{1}) + O((log P)^{−1/3})} ,
(331)  𝔗^{−} ≥ 𝔑 {f(s_{1}) + O((log P)^{−1/3})} ,

where F and f are the functions of the linear sieve (see Iwaniec [12], [13]), s_{1} is specified by (298),

𝔑 = ∏_{z_{0}≤p<z_{1}} (1 − \frac{ψ(p)}{p})

and ψ is defined by (326). We use (10), (11), (292), (294) and (326) to find that (332) 𝔑 ≫ \frac{log z_{0}}{log z_{1}} ≫ (log P)^{−1} (log log P) .

We choose α = 0.00983 and use that if γ denotes the Euler constant and s ∈ (2, 3) then F (s) = 2e^{γ}s^{−1} and f(s) = 2e^{γ}s^{−1} log(s − 1). We find that if s_{1} is specified by (298) and ε is sufficiently small then 3f(s_{1}) − 2F (s_{1}) > 10^{−4} . (333) From (329) – (333) we obtain

(334) W_{2} ≥ 𝔑^{3}(3f(s_{1}) − 2F (s_{1}) + O((log P)^{−1/3}) ) ≫ (log P)^{−3} (log log P)^{3} . We use (27), (294), (311), (320), (322), (323), (325), (327), (328) and (334)

and we find that if α = 0.00983 then the estimate (293) holds. It remains to notice that this α satisfies 101 < α^{−1} < 102 and the proof of Theorem 1 is complete.

□

5 The proof of Theorem 2.

We recall that in this section bold style letters denote four–dimensional vectors. We also assume that the components of d are always squarefree. Now we define

(335)  S_{d}(q, m, n) = ∏_{i=1}^{4} S(q, md_{i}^{2}, n_{i}) ,  S_{d}(q, m) = S_{d}(q, m, 0) ,

(336)  I_{d}(β, u) = ∏_{i=1}^{4} I(β, u_{i}d_{i}^{−1}) ,  f_{d}(α) = ∏_{i=1}^{4} f_{d_{i}}(α)

and let

T_{N}(q, d) = ((q, N)(q, d_{1}^{2}) … (q, d_{4}^{2}))^{1/2} .

One of the main points in [1] is the estimation of the sum

V (q, d, n, v, N) = ∑_{a(q)^{∗}} e_{q}(v\overline{a} − Na) S_{d}(q, a, n) . (337)

In Lemma 1 of [1] Brüdern and Fouvry use estimates for Kloosterman and Salié sums and prove the inequality

V (q, d, n, v, N) ≪ q^{5/2+ε} T_{N}(q, d) . (338)

From (338) follows, in particular, that the singular series

Σ_{1}(d, N) = ∑_{q=1}^{∞} q^{−4} V (q, d, 0, 0, N) (339)

is absolutely convergent. The arithmetic properties of the function Σ_{1}(d, N) are studied in detail in Section 2.4 of [1]. Define

B(d, N) = ∑_{\begin{matrix}x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2}=N \\ x_{i}≡0 (d_{i}) \end{matrix}} ω(x_{1}) … ω(x_{4})

and

R(d, N) = B(d, N) − κ_{1} P^{2} \frac{Σ_{1}(d, N)}{d_{1}d_{2}d_{3}d_{4}} , (340)

where ω, κ_{1} and Σ_{1} are specified by (18), (24) and (339), respectively. For any positive D we define

H^{∗}(D) = ∑_{(D^{∗})} |R(d, N)| , (341)

where ∑_{(D^{∗})} means that the summation is taken over squarefree odd integers d_{1}, d_{2}, d_{3}, d_{4} ≤ D. We prove the following Proposition 3. Suppose that D ≤ P^{1/8−10ε}. Then we have

H^{∗}(D) ≪ P^{2−ε} .

This statement is an analog of Theorem 3 from [1]. In this paper Brüdern and Fouvry use the approach of Estermann [4] and their upper bound for d_{i} is P^{θ}, where θ < 1/11. Here we use weighted exponential sums and apply Lemma 12 and our result becomes stronger.

Applying the arguments from Section 3 of [1] we can see that our Theorem 2 is a consequence of Proposition 3.

Proof of Proposition 3. It is enough to establish the inequality

H^{∗}(D) ≪ P^{3/2+10ε} D^{4} . (342)

We use again the Kloosterman method. It is clear that

(343)  B(d, N) = ∫_{0}^{1} e(−Nα)f_{d}(α) dα

= ∑_{q≤P} ∑_{a(q)^{∗}} ∫_{B(q,a)} e( − N(\frac{a}{q} + β) ) f_{d}(\frac{a}{q} + β) dβ ,

where the set of integration B(q, a) and the function f_{d} are defined by (54) and (336), respectively. We use (335), (336) and Lemma 12 to represent the integrand from the formula above in the form

\frac{P^{4} e( − N(\frac{a}{q} + β) )}{q^{4} d_{1}d_{2}d_{3}d_{4}} ∑_{\begin{matrix}n∈ℤ^{4} \\ |n_{i}|≤d_{i}P^{ε} \end{matrix}} S_{d}(q, a, n) I_{d}(βN, −P q^{−1}n) + O(P^{−A}) .

We substitute this expression for the integrand in (343) and change the variable of integration to get

B(d, N) = \frac{P^{2}}{d_{1}d_{2}d_{3}d_{4}} ∑_{q≤P} q^{−4} ∑_{\begin{matrix}n∈ℤ^{4} \\ |n_{i}|≤d_{i}P^{ε} \end{matrix}} ∑_{a(q)^{∗}} ∫_{B′(q,a)} e_{q}(−Na) e(−β)

× S_{d}(q, a, n) I_{d}(β, −P q^{−1}n) dβ + O(P^{−A}) ,

where the set B′(q, a) is defined by (62). Then we change the order of integration and summation over a to find that

(344) B(d, N) = \frac{P^{2}}{d_{1}d_{2}d_{3}d_{4}} ∑_{q≤P} q^{−4} ∑_{\begin{matrix}n∈ℤ^{4} \\ |n_{i}|≤d_{i}P^{ε} \end{matrix}} ∫_{|β|≤ \frac{P}{q}} e(−β)I_{d}(β, −P q^{−1}n)

× ∑_{\begin{matrix}a(q)^{∗} \\ \overline{a}∈A(q,β) \end{matrix}} e_{q}(−Na) S_{d}(q, a, n) dβ + O(P^{−A}) ,

where A(q, β) is the set of residue classes modulo q, whose properties were described in Section 3.4.2. In particular, there exists a function σ(v, q, β), satisfying (99) and such that

∑_{\begin{matrix}a(q)^{∗} \\ \overline{a}∈A(q,β) \end{matrix}} … = ∑_{−q/2<v≤q/2} σ(v, q, β) ∑_{a(q)^{∗}} e_{q}(\overline{a}v) …, (345)

We represent B(d, N) in the form

B(d, N) = B_{1} + B_{2} + O(P^{−A}) , (346)

where in B_{1} the integration is taken over β such that |β| ≤ P (2q)^{−1}, whilst in B_{2} we integrate over β satisfying P (2q)^{−1} < |β| ≤ P q^{−1}.

Consider B_{2}. From (99), (337), (344) and (345) we obtain

(347)  B_{2} = \frac{P^{2}}{d_{1}d_{2}d_{3}d_{4}} ∑_{q≤P} q^{−4} ∑_{\begin{matrix}n∈ℤ^{4} \\ |n_{i}|≤d_{i}P^{ε} \end{matrix}} ∫_{\frac{P}{2q} ≤|β|≤ \frac{P}{q}} I_{d}(β, −P q^{−1}n)

× e(−β) ∑_{−q/2<v≤q/2} σ(v, q, β) V (q, d, n, v, N) dβ

≪ \frac{P^{2}}{d_{1}d_{2}d_{3}d_{4}} ∑_{q≤P} q^{−4} ∑_{\begin{matrix}n∈ℤ^{4} \\ |n_{i}|≤d_{i}P^{ε} \end{matrix}} ∑_{|v|≤P}(1 + |v|)^{−1}

× |V (q, d, n, v, N)| ∫ ∞_{\frac{P}{2q}} |I_{d}(β, −P q^{−1}n)|dβ .

Applying (336) and Lemma 9 (iii) we find that

∫ ∞_{\frac{P}{2q}} |I_{d}(β, −P q^{−1}n)|dβ ≪ ∫ ∞_{\frac{P}{2q}} β^{−2}dβ ≪ qP^{−1} (348)

and from (338), (347) and (348) we obtain

B_{2} ≪ P^{1+6ε} ∑_{q≤P} q^{−1/2} T_{N}(q, d) . (349)

Consider now B_{1}. In this case the set A(q, β) is a complete system of residues modulo q and we have

B_{1} = \frac{P^{2}}{d_{1}d_{2}d_{3}d_{4}} ∑_{q≤P} q^{−4} ∑_{\begin{matrix}n∈ℤ^{4} \\ |n_{i}|≤d_{i}P^{ε} \end{matrix}} J(q, d, n, N) V (q, d, n, 0, N) , (350)

where

J(q, d, n, N) = ∫_{|β|≤ \frac{P}{2q}} e(−β)I_{d}(β, −P q^{−1}n)dβ .

Using (24), (336), (348) and Lemma 10 (ii) we find that

(351) J(q, d, n, N) = \begin{cases}κ_{1} + O(qP^{−1}) & if n = 0 , \\ O(qP^{−1+2ε}( ∑_{i=1}^{4} |n_{i}| d_{i}^{−1})^{−1} ) & otherwise . \end{cases}

From (338), (339), (350) and (351) we obtain

B_{1} = κ_{1}P^{2} \frac{Σ_{1}(d, N)}{d_{1}d_{2}d_{3}d_{4}} + O(B_{3}) + O(B_{4}) + O(B_{5}) , (352)

where

B_{3} = \frac{P^{2}}{d_{1}d_{2}d_{3}d_{4}} ∑_{q>P} q^{−3/2+ε} T_{N}(q, d) ,
B_{4} = \frac{P^{1+3ε}}{d_{1}d_{2}d_{3}d_{4}} ∑_{q≤P} q^{−1/2} T_{N}(q, d) ,
B_{5} = \frac{P^{1+3ε}}{d_{1}d_{2}d_{3}d_{4}} ∑_{q≤P} q^{−1/2} T_{N}(q, d) ∑_{\begin{matrix}0≠n∈ℤ^{4} \\ |n_{i}|≤d_{i}P^{ε} \end{matrix}} ( ∑_{i=1}^{4} |n_{i}| d_{i}^{−1} )^{−1} .

Using the last three formulas as well as (349) we can establish that

∑_{(D^{∗})} |B_{i}| ≪ P^{3/2+10ε}D^{4} for i = 2, 3, 4, 5 . (353)

The calculations are not difficult and we leave them to the reader.

We take into account (340), (341), (346), (352) and (353) and prove that the estimate (342) is correct. This completes the proof of Proposition 3.

□

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Mathematical Institute 24–29 St. Giles’ Oxford OX1 3LB England Email: rhb@maths.ox.ac.uk

Department of Mathematics Plovdiv University “P. Hilendarski” 24 “Tsar Asen” str. Plovdiv 4000 Bulgaria Email: dtolev@pu.acad.bg