Lagrange’s four squares theorem with variables of special type

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.

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 conjecture has not been proved so far. We should mention, however, that Hua [6] proved that all large integers N such that N ≡ 5 (mod 24), are sums of five squares of primes.

Greaves [3], Plaksin [10] and Shields [11] established the solvability of (1) with two prime and two integer variables provided that N is sufficiently large and satisfies a natural congruence condition. Brüdern and Fouvry [1] proved that any sufficiently large integer N ≡ 4 (mod 24), can be represented in the form (1), where each variable is a number of type P_{34} (as usual, by P_{r} we denote any integer having at most r prime factors, counted according to multiplicity).

D.R. Heath–Brown and the author considered recently the equation (1) with more restrictive conditions on the variables than in the paper of Brüdern and Fouvry. In [5] two theorems were proved. The first of them states that every sufficiently large N ≡ 4 (mod 24) can be represented in the form (1), where x_{1} is a prime and where each of x_{2}, x_{3}, x_{4} is a number of type P_{101}. The most important part of the proof of this result is the establishment of Propositions 1 and 2. The second of them asserts, that the sum L(k, N), defined by (3), can be approximated in some average sense to the expected main term, which we can find by a formal application of the circle method. After that the proof of Theorem 1 of [5] can be established by using the vector sieve. This sieve method was proposed by Iwaniec [7] and was also used by Brüdern and Fouvry [1], [2] and by the author [12] – [14].

The second theorem of [5] states that for every sufficiently large N ≡ 4 (mod 24) the equation (1) is solvable in variables of type P_{25}. To prove this result we use Proposition 3 of [5]. It, roughly speaking, states that the number of the solutions of (1) in integers lying in progressions, can be approximated on average by the expected value with better level of distribution than in the relevant theorem from the paper [1]. Then the result follows again by application of the vector sieve.

The aim of the present paper is to show that improvements upon the results of [5] (of about 20%, in some sense) can be achieved by attaching Kuhn’s weight to one of the variables. The following theorems hold:

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

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

where x_{1} = P_{78}, x_{2} = P_{80}, x_{3} = P′_{80} and where q is a prime.

Theorem 2. Every sufficiently large integer N, satisfying N ≡ 4 (mod 24), can be represented in the form (1), where each of the variables is of type P_{21}.

We present only the proof of Theorem 1. Theorem 2 can be proved in the same way. We omit the calculations because they are similar to those in the relevant parts of [5], [12] – [14].

Acknowledgements. This paper was written during the author’s stay at Max–Planck–Institut für Mathematik in the framework of the “Special activity on Analytic Number Theory”. The author thanks for financial support and for the excellent working conditions.

2 Notations.

We assume that the integer N is sufficiently large and satisfies the congruence N ≡ 4 (mod 24). Denote P = N^{1/2}. Let ε ∈ (0, 10^{−6}) be arbitrarily small and let A > 10^{6} be arbitrarily large number (they may not be the same in different occurrences). The constants in O–terms and ≪–symbols are absolute or depend on ε and A. For positive U and V we write U ≍ V as an abbreviation of U ≪ V ≪ U. The letters p and q are reserved for prime numbers. As usual, μ(n), τ(n), Ω(n) denote, respectively, the Möbius function, the number of divisors of n and the number of prime factors of n, counted according to multiplicity. 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 by (m_{1}, m_{2}) and [m_{1}, m_{2}] the greatest common divisor and, respectively, the least common multiple of the integers of m_{1}, m_{2}. However, if u, v are real numbers then (u, v) means the interval with endpoints u and v. Finally, by bold style letters we denote three–dimensional vectors in the following way: d = ⟨d_{1}, d_{2}, d_{3}⟩. The meaning is always clear from the context.

3 Proposition.

First we state a proposition, which is an analog of Proposition 2 of [5]. 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}

and denote ω(x) = ∏_{i=1}^{3} ω_{0}(x_{i}P^{−1}). For any vector d with squarefree odd components we consider the sum

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

In order to prove Theorem 1 we have to approximate this sum by another expression, which is easier to work with. The most difficult part of the proof is to establish that this expression approximates L(d, N) in some average sense indeed. We have the following:

Proposition. Suppose that K_{1}, K_{2}, K_{3} are positive numbers, satisfying

K_{1} ≤ K_{2}K_{3} , K_{2} ≤ K_{1}K_{3} , K_{3} ≤ K_{1}K_{2} , K_{1}K_{2}K_{3} ≤ P^{2/23−ε} .

Let β_{i}(d) be real functions, supported on the set of positive squarefree odd integers, and such that

|β_{i}(d)| ≤ τ^{2}(d) ,  β_{i}(d) = 0 if d > K_{i} ,  i = 1, 2, 3 .

There exist quantities N_{0} = N_{0}(N), ξ_{0} = ξ_{0}(N), R(d, N) with the following properties:

For N_{0} and ξ_{0} we have

(4)  N_{0} ≍ \frac{P^{2}}{log P} ,  1 ≪ ξ_{0} ≪ log log P .

The quantity R is defined for vectors d with squarefree odd components and can be decomposed in the following way

R(d, N) = ∏_{p∥d_{1}d_{2}d_{3}} ψ_{1}(p, N) ∏_{p^{2}∥d_{1}d_{2}d_{3}} ψ_{2}(p, N) ∏_{p^{3}∥d_{1}d_{2}d_{3}} ψ_{3}(p, N) , (5)

where ψ_{i}(p, N) , i = 1, 2, 3, are functions defined for primes p > 2 and such that

(6) ψ_{1}(3, N) = 1/3 , 1/4 ≤ ψ_{1}(p, N) ≤ 4 , ψ_{1}(p, N) = 1 + O(p^{−1}) ,
0 ≤ ψ_{i}(p, N) ≤ 4 ,  i = 2, 3 .

We also have

R(d, N) ≪ τ^{2}(d_{1})τ^{2}(d_{2})τ^{2}(d_{3}) . (7)

Furthermore, if 2 ∤ k_{i}l_{j} and μ(k_{i}l_{j}) ≠ 0 for 1 ≤ i, j ≤ 3 then

R(⟨k_{1}l_{1}, k_{2}l_{2}, k_{3}l_{3}⟩, N) = R(k, N) R(l, N) . (8)

Finally, the following estimate holds:

∑_{d_{1},d_{2},d_{3}} β_{1}(d_{1})β_{2}(d_{2})β_{3}(d_{3})(L(d, N) − \frac{N_{0} ξ_{0} R(d, N)}{d_{1}d_{2}d_{3}}) ≪ P^{2}(log P)^{−A} . (9)

Let us notice, that the terms N_{0} and ξ_{0}R(d, N)/(d_{1}d_{2}d_{3}) can be considered as the “singular integral” and, respectively, the “singular series”, for the sum L(d, N). Hence the second term in the brackets in (9) is the expected approximation to L(d, N), which we can find by a formal application of the circle method. For the exact definitions of N_{0}, ξ_{0} and R(d, N) we refer the reader to [5].

The present proposition is a straightforward generalization of Proposition 2 from [5] (where we have K_{1} = K_{2} = K_{3}). The proof is long and complicated, but differs slightly from the proof of the corresponding assertion in [5], so we omit it. We also notice, that in Proposition 2 from [5] the “singular series” is given in the form of series indeed, and its decomposition as a product and the properties of the factors are established in [5], Section 4.

4 Proof of Theorem 1.

4.1 Beginning of the proof.

Suppose that α_{1}, α_{2}, α_{3} and θ are constants, such that

0 < θ < 1 ,  0 < α_{1} < α_{3} < 1 ,  α_{1} < α_{2} < 1 . (10)

Define

z_{0} = (log P)^{1000} ,  z_{i} = P^{α_{i}} ,  i = 1, 2, 3 , (11)

and let

𝔓_{0} = ∏_{2<p<z_{0}} p ,  𝔓_{i} = ∏_{z_{0}≤p<z_{i}} p ,  i = 1, 2 . (12)

Consider the sum

(13)  Γ = ∑_{\begin{matrix}q^{2}+x_{1}^{2}+x_{2}^{2}+x_{3}^{2}=N \\ (x_{1},𝔓_{0}𝔓_{1})=(x_{2}x_{3},𝔓_{0}𝔓_{2})=1 \end{matrix}} ω(x) (1 − θ ∑_{\begin{matrix}z_{1}≤p<z_{3} \\ p|x_{1} \end{matrix}} (1 − \frac{log p}{log z_{3}} ) ) .

Using the condition N ≡ 4 (24) and the definition of ω(x) we find that the solutions of (2), such that 2 | qx_{1}x_{2}x_{3}, are not counted in Γ.

Our aim is to show that for suitable constants α_{1}, α_{2}, α_{3}, θ, satisfying (10), we have

Γ ≫ P^{2}(log P)^{−4} . (14)

Having proved this, consider the part Γ′ of Γ, consisting of all terms, such that x_{1} ≡ 0 (p^{2}) for some prime p ∈ [z_{1}, z_{3}). Using (10) and (11) we get

(15) Γ′ ≪ ∑_{z_{1}≤p<z_{3}} ∑_{\begin{matrix}q^{2}+x_{1}^{2}+x_{2}^{2}+x_{3}^{2}=N \\ x_{1}≡0(p^{2}) \end{matrix}} 1 = ∑_{z_{1}≤p<z_{3}} ∑_{k≤N} ( ∑_{q^{2}+x_{3}^{2}=k} 1) ∑_{\begin{matrix}x_{1}^{2}+x_{2}^{2}=N−k \\ x_{1}≡0(p^{2}) \end{matrix}} 1

≪ P^{ε} ∑_{z_{1}≤p<z_{3}} ∑_{\begin{matrix}x_{1},x_{2}≤P \\ x_{1}≡0(p^{2}) \end{matrix}} 1 ≪ P^{1+ε}(P z_{1}^{−1} + z_{3}) ≪ P^{2−ε} .

From (13) – (15) we conclude that there are ≫ P^{2}(log P)^{−4} quadruples q, x_{1}, x_{2}, x_{3}, satisfying (2) and such that q is a prime, x_{1} has no multiple prime factors p ∈ [z_{1}, z_{3}),

(x_{1}, 2𝔓_{0}𝔓_{1}) = (x_{2}x_{3}, 2𝔓_{0}𝔓_{2}) = 1 (16)

and

1 − θ ∑_{\begin{matrix}z_{1}≤p<z_{3} \\ p|x_{1} \end{matrix}} (1 − \frac{log p}{log z_{3}} ) > 0 . (17)

From (11) and (16) we get

Ω(x_{2}), Ω(x_{3}) ≤ α_{2}^{−1} . (18)

Furthermore, from (16) and (17) we easily obtain

Ω(x_{1}) < θ^{−1} + α_{3}^{−1} . (19)

The arguments are similar to those in [4], Chapter 9, § 2, for example, so we omit them.

We see that our aim is to choose constants α_{1}, α_{2}, α_{3}, θ, satisfying (10), in such a way, that the estimate (14) holds, and at the same time the number

max (α_{2}^{−1}, θ^{−1} + α_{3}^{−1})

should be as small as possible.

It is clear that

Γ = F − θ G , (20) where

(21)  F = ∑_{\begin{matrix}q^{2}+x_{1}^{2}+x_{2}^{2}+x_{3}^{2}=N \\ (x_{1},𝔓_{0}𝔓_{1})=(x_{2}x_{3},𝔓_{0}𝔓_{2})=1 \end{matrix}} ω(x) ,

(22)  G = ∑_{z_{1}≤p<z_{3}} (1 − \frac{log p}{log z_{3}} ) ∑_{\begin{matrix}q^{2}+x_{1}^{2}+x_{2}^{2}+x_{3}^{2}=N \\ (x_{1},𝔓_{0}𝔓_{1})=(x_{2}x_{3},𝔓_{0}𝔓_{2})=1 \\ x_{1}≡0 (p) \end{matrix}} ω(x) .

In order to prove (14) we have to estimate F from below and G from above.

4.2 The estimation of F.

To study the sum F, defined by (21), we apply the vector sieve. We proceed exactly as in Section 4 of [5]. For reader’s convenience we present the main points, but omit the calculations.

Using the fundamental property of the Möbius function we represent F in the form

F = ∑_{q^{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} , (23)

where

Φ_{i} = ∑_{d|(x_{i},𝔓_{0})}μ(d) , i = 1, 2, 3 ; (24)

Λ_{1} = ∑_{d|(x_{1},𝔓_{1})}μ(d) ;  Λ_{i} = ∑_{d|(x_{i},𝔓_{2})}μ(d) ,  i = 2, 3 . (25)

Define

D_{0} = P^{ε};  D_{i} = P^{η_{i}} ,  i = 1, 2 , (26)

where η_{1}, η_{2} are constants such that

2 ≤ s_{2} = \frac{η_{2}}{α_{2}} ≤ 3 ≤ s_{1} = \frac{η_{1}}{α_{1}} ≤ 4 , α_{1} + α_{3} ≤ η_{1} . (27)

Let λ_{i}^{±} , i = 0, 1, 2, be the Rosser functions of orders D_{i}, respectively. Denote

Φ_{i}^{±} = ∑_{d|(x_{i},𝔓_{0})}λ_{0}^{±}(d) , i = 1, 2, 3 ; (28)

Λ_{1}^{±} = ∑_{d|(x_{1},𝔓_{1})}λ_{1}^{±}(d) ;  Λ_{i} = ∑_{d|(x_{i},𝔓_{2})}λ_{2}^{±}(d) ,  i = 2, 3 . (29)

The definition and the properties of the Rosser weights can be found in Iwaniec [8], [9]. In particular, we have

|λ_{i}^{±}(d)| ≤ 1 , λ_{i}^{±}(d) = 0 if μ(d) = 0 or d > D_{i} , i = 0, 1, 2 ; (30)

Φ_{i}^{−} ≤ Φ_{i} ≤ Φ_{i}^{+} ,  Λ_{i}^{−} ≤ Λ_{i} ≤ Λ_{i}^{+} ,  i = 1, 2, 3 . (31)

The lower estimate for F is based on the elementary inequality

(32)  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}^{−}
−5 Φ_{1}^{+}Φ_{2}^{+}Φ_{3}^{+}Λ_{1}^{+}Λ_{2}^{+}Λ_{3}^{+} ,

which proof is similar to the proof of Lemma 13 of [1]. From (23) and (32) we get

F ≥ F_{1} + ⋯ + F_{6} − 5F_{7} , (33)

where F_{i} are the contributions arising from the consecutive terms from the right–hand side of (32).

Consider, for example, F_{1}. Using (28), (29) and changing the order of summation we write it in the form

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

where

(34)  β_{1}(d) = ∑_{\begin{matrix}k|𝔓_{0} , l|𝔓_{1} \\ kl=d \end{matrix}} λ_{0}^{−}(k) λ_{1}^{+}(l) ,

(35)  β_{i}(d) = ∑_{\begin{matrix}k|𝔓_{0} , l|𝔓_{2} \\ kl=d \end{matrix}} λ_{0}^{+}(k) λ_{2}^{+}(l) , i = 2, 3 .

From this point onwards we assume that η_{1} < 2η_{2} , η_{1} + 2η_{2} < 2/23 . (36) We take ε sufficiently small and use (26), (30) and (36) to verify that the functions β_{i}(d) satisfy the requirements of our Proposition (with K_{1} = D_{0}D_{1}, K_{2} = K_{3} = D_{0}D_{2}). Hence we have F_{1} = N_{0}ξ_{0}F′_{1} + O(P^{2}(log P)^{−A}) (37) where

F′_{1} = ∑_{d_{1},d_{2},d_{3}} \frac{β_{1}(d_{1}) β_{2}(d_{2}) β_{3}(d_{3})}{d_{1}d_{2}d_{3}}R(d, N) . (38)

From (8), (34), (35) and (38) we get F′_{1} = H^{−}V^{+} , (39) where

(40)  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) ,
(41)  V^{+} = ∑_{l_{1}|𝔓_{1} ; l_{2},l_{3}|𝔓_{2}} \frac{λ_{1}^{+}(l_{1})λ_{2}^{+}(l_{2})λ_{2}^{+}(l_{3})}{l_{1}l_{2}l_{3}} R(l, N) .

From (37) and (39) an asymptotic formula for F_{1} follows.

We proceed with F_{i} , 2 ≤ i ≤ 7 in the same way and use (33) to find F ≥ N_{0}ξ_{0}(3H^{−}V^{+} + H^{+}V_{1}^{−} + 2H^{+}V_{2}^{−} − 5H^{+}V^{+}) + O(P^{2}(log P)^{−A}) , (42) where

(43)  V_{1}^{−} = ∑_{l_{1}|𝔓_{1} ; l_{2},l_{3}|𝔓_{2}} \frac{λ_{1}^{−}(l_{1})λ_{2}^{+}(l_{2})λ_{2}^{+}(l_{3})}{l_{1}l_{2}l_{3}} R(l, N) ,
(44)  V_{2}^{−} = ∑_{l_{1}|𝔓_{1} ; l_{2},l_{3}|𝔓_{2}} \frac{λ_{1}^{+}(l_{1})λ_{2}^{−}(l_{2})λ_{2}^{+}(l_{3})}{l_{1}l_{2}l_{3}} R(l, N) .

We note that from (7) and (30) follows V^{+}, V_{1}^{−}, V_{2}^{−} ≪ (log P)^{12} . (45)

Consider H^{±}. Obviously s_{0} = log D_{0}/ log z_{0} → ∞ as P → ∞. Hence we may expect that the Rosser functions λ_{0}^{±} behave like the Möbius function. Indeed, if we substitute μ(k_{1})μ(k_{2})μ(k_{3}) for λ_{0}^{±}(k_{1})λ_{0}^{+}(k_{2})λ_{0}^{+}(k_{3}) and denote the new sum by H_{0} then we can find that H^{±} = H_{0} + O(exp(−\sqrt{log P})) . (46) A detailed proof of this asymptotic formula is available in Section 3.7 of [14].

Furthermore, it is easy to prove that H_{0} ≍ (log log P)^{−3} . (47)

From (4), (42) and (45) – (47) we get F ≥ N_{0}ξ_{0}H_{0} (V_{1}^{−} + 2V_{2}^{−} − 2V^{+}) + O(P^{2}(log P)^{−A}) . (48)

4.3 The estimation of G.

We write the sum G, defined by (22), in the form

(49)  G = ∑_{z_{1}≤p<z_{3}} (1 − \frac{log p}{log z_{3}} ) ∑_{\begin{matrix}q^{2}+x_{1}^{2}+x_{2}^{2}+x_{3}^{2}=N \\ x_{1}≡0 (p) \end{matrix}} ω(x) Φ_{1}Φ_{2}Φ_{3}Λ_{1}Λ_{2}Λ_{3} ,

where Φ_{i}, Λ_{i} are specified by (24) and (25).

For any prime p ∈ [z_{1}, z_{3}) consider the upper Rosser function λ^{(p)} of order D_{1}/p, where D_{1} is given by (26). Let

Λ^{(p)} = ∑_{d|(x_{1},𝔓_{1})}λ^{(p)}(d) . (50)

We have |λ^{(p)}(d)| ≤ 1 , λ^{(p)}(d) = 0 if μ(d) = 0 or d > D_{1}/p (51) and also Λ_{1} ≤ Λ^{(p)} . (52)

Using (31), (49) and (52) we find that G ≤ G_{1} , (53) where

(54) G_{1} = ∑_{z_{1}≤p<z_{3}} (1 − \frac{log p}{log z_{3}} ) ∑_{\begin{matrix}q^{2}+x_{1}^{2}+x_{2}^{2}+x_{3}^{2}=N \\ x_{1}≡0 (p) \end{matrix}} ω(x) Φ_{1}^{+}Φ_{2}^{+}Φ_{3}^{+}Λ^{(p)}Λ_{2}^{+}Λ_{3}^{+} .

From (28), (29), (50) and (54) after some rearrangements we obtain

G_{1} = ∑_{d_{1},d_{2},d_{3}} β_{1}^{∗}(d_{1}) β_{2}(d_{2}) β_{3}(d_{3}) L(d, N) ,

where

β_{1}^{∗}(d) = ∑_{\begin{matrix}z_{1}≤p<z_{3} \\ k|𝔓_{0} , l|𝔓_{1} \\ klp=d \end{matrix}} (1 − \frac{log p}{log z_{3}} ) λ_{0}^{+}(k) λ^{(p)}(l) (55)

and where β_{i}(d) , i = 2, 3, are defined by (35).

From (30), (36) and (51) follows that we can apply our Proposition again (with K_{1} = D_{0}D_{1}, K_{2} = K_{3} = D_{0}D_{2}) and we find G_{1} = N_{0}ξ_{0}G_{2} + O(P^{2}(log P)^{−A}) , (56) where

G_{2} = ∑_{d_{1},d_{2},d_{3}} \frac{β_{1}^{∗}(d_{1})β_{2}(d_{2})β_{3}(d_{3})}{d_{1}d_{2}d_{3}} R(d, N) . (57)

From (5), (8), (35), (55) and (57) we get G_{2} = H^{+}G_{3} , (58) where H^{+} is defined by (40),

G_{3} = ∑_{z_{1}≤p<z_{3}} \frac{ψ_{1}(p, N)}{p}(1 − \frac{log p}{log z_{3}} ) V^{(p)} (59)

and where

V^{(p)} = ∑_{l_{1}|𝔓_{1} ; l_{2},l_{3}|𝔓_{2}} \frac{λ^{(p)}(l_{1})λ_{2}^{+}(l_{2})λ_{2}^{+}(l_{3})}{l_{1}l_{2}l_{3}} R(l, N) . (60)

From (7), (30) and (51) we get V^{(p)} ≪ (log P)^{12} . (61) Furthermore, using (6), (11), (59) and (61) we find that G_{3} = G_{4} + O(P^{−ε}) , (62) where

G_{4} = ∑_{z_{1}≤p<z_{3}} \frac{1}{p}(1 − \frac{log p}{log z_{3}} ) V^{(p)} . (63)

Finally, from (4), (46), (47), (53), (56), (58), (61) and (62) we obtain G ≤ N_{0}ξ_{0}H_{0}G_{4} + O(P^{2}(log P)^{−A})) . (64)

4.4 The lower bound for Γ and the end of the proof.

We take into account (20), (48) and (64) to get Γ ≥ N_{0}ξ_{0}H_{0}M + O(P^{2}(log P)^{−A})) , (65) where M = V_{1}^{−} + 2V_{2}^{−} − 2V^{+} − θG_{4} . (66) We shall find approximations for the expressions V^{+}, V_{1}^{−}, V_{2}^{−} and V^{(p)}, defined, respectively, by (41), (43), (44) and (60).

Consider, for example, V^{+}. Suppose that l_{1} | 𝔓_{1} and l_{2}, l_{3} | 𝔓_{2}. Using (12) we find that the condition (l_{i}, l_{j}) > 1 implies (l_{i}, l_{j}) ≥ z_{0}. From this fact and (7) we easily obtain V^{+} = Ṽ^{+} + O((log P)^{−100}) , (67) where in Ṽ^{+} the summation is restricted to l_{i} such that (l_{1}, l_{2}) = (l_{1}, l_{3}) = (l_{2}, l_{3}) = 1. In this case R(l, N) = ψ_{1}(l_{1}, N)ψ_{1}(l_{2}, N)ψ_{1}(l_{3}, N), where we have defined ψ_{1}(l, N) = ∏_{p|l} ψ_{1}(p, N). Using the fundamental property of the Möbius function and changing the order of summation we find that

(68) Ṽ^{+} = ∑_{h_{1},h_{2},h_{3}|𝔓_{2}} μ(h_{1})μ(h_{2})μ(h_{3})

× ∑_{\begin{matrix}l_{1}|𝔓_{1} , l_{2},l_{3}|𝔓_{2} \\ l_{1}≡0 ([h_{2},h_{3}]) \\ l_{2}≡0 ([h_{1},h_{3}]) \\ l_{3}≡0 ([h_{1},h_{2}]) \end{matrix}} \frac{λ_{1}^{+}(l_{1})λ_{2}^{+}(l_{2})λ_{2}^{+}(l_{3})}{l_{1}l_{2}l_{3}}ψ_{1}(l_{1}, N)ψ_{1}(l_{2}, N)ψ_{1}(l_{3}, N) .

If h_{i} | 𝔓_{2} and h_{i} > 1 then h_{i} ≥ z_{0}. So, after some standard calculations, which we leave to the reader, we get

Ṽ^{+} = \overline{V^{+}} + O((log P)^{−100}) , (69)

where \overline{V^{+}} is the contribution of the terms with h_{1} = h_{2} = h_{3} = 1. From (67),

(69) and from the definition of \overline{V^{+}} we obtain

V^{+} = T_{1}^{+} (T_{2}^{+})^{2} + O((log P)^{−100}) , (70)

where

T_{i}^{±} = ∑_{l|𝔓_{i}} \frac{λ_{i}^{±}(l)}{l}ψ_{1}(l, N) , i = 1, 2 . (71)

Similarly we find that

(72)  V_{1}^{−} = T_{1}^{−} (T_{2}^{+})^{2} + O((log P)^{−100}) ,
(73)  V_{2}^{−} = T_{1}^{+} T_{2}^{−} T_{2}^{+} + O((log P)^{−100}) ,
(74)  V^{(p)} = T^{(p)} (T_{2}^{+})^{2} + O((log P)^{−100}) ,

where

T^{(p)} = ∑_{l|𝔓_{1}} \frac{λ^{(p)}(l)}{l}ψ_{1}(l, N) . (75)

From (63), (66), (70) and (72) – (74) we get M = T_{2}^{+} M^{∗} + O((log P)^{−100}) , (76) where

M^{∗} = T_{1}^{−}T_{2}^{+} + 2T_{1}^{+}T_{2}^{−} − 2T_{1}^{+}T_{2}^{+} − θT_{2}^{+}G_{5} ,

and

G_{5} = ∑_{z_{1}≤p<z_{3}} \frac{1}{p}(1 − \frac{log p}{log z_{3}} ) T^{(p)} . (77)

We write M^{∗} in the form M^{∗} = M_{1} T_{2}^{+} + 2 M_{2} T_{1}^{+} , (78) where M_{1} = T_{1}^{−} − 2θ_{1}T_{1}^{+} − θG_{5} , M_{2} = T_{2}^{−} − θ_{2}T_{2}^{+} (79) and where θ_{1}, θ_{2} > 0 are constants such that θ_{1} + θ_{2} = 1 . (80)

Suppose that F (s) and f(s) are the functions of the linear sieve. We have

(81) f(s) = 2e^{γ}s^{−1} log(s − 1)  for  2 ≤ s ≤ 4 ;
(82) F (s) = \begin{cases}2e^{γ}s^{−1} & for  1 ≤ s ≤ 3 , \\ 2e^{γ}s^{−1}(1 + ∫_{2}^{s−1} t^{−1} log(t − 1) dt) & for  3 ≤ s ≤ 4 . \end{cases}

Here γ = 0.577 … is the Euler constant.

Denote

𝔑_{i} = ∏_{z_{0}≤p<z_{i}} (1 − \frac{ψ_{1}(p, N)}{p}) , i = 1, 2 . (83)

Using (6) and (11) we get

(84)  𝔑_{i} ≍ \frac{log log P}{log P} ,  i = 1, 2 .

From the theory of the linear sieve (see Iwaniec [8], [9]) follows that if the conditions (10), (11), (26) and (27) hold and if p ∈ [z_{1}, z_{3}) then the quantities T_{i}^{±} and T^{(p)}, defined by (71) and (75), satisfy

(85)  𝔑_{i} ≤ T_{i}^{+} ≤ 𝔑_{i} {F (s_{i}) + O((log P)^{−1/3}))} ,  i = 1, 2 ,
(86)  T_{i}^{−} ≥ 𝔑_{i} {f(s_{i}) + O((log P)^{−1/3}))} ,  i = 1, 2 ,
(87)  T^{(p)} ≤ 𝔑_{1} \{F (\frac{log D_{1}/p}{log z_{1}}) + O((log P)^{−1/3}))\} .

Consider the sum G_{5}, defined by (77). From (87) we get G_{5} ≤ 𝔑_{1} {G_{6} + O((log P)^{−1/4}))} , (88) where

G_{6} = ∑_{z_{1}≤p<z_{3}} \frac{1}{p} (1 − \frac{log p}{log z_{3}} ) F (\frac{log D_{1}/p}{log z_{1}}) .

It follows from (27) that the argument of the function F (s) in the last formula belongs to the segment [1, 3], so we can apply (82). Using Abel’s summation formula and the Prime Number Theorem after some calculations we obtain G_{6} = 2e^{γ} κ + O((log P)^{−1}) , (89) where

κ = κ(α_{1}, α_{3}, s_{1}) = ∫_{α_{3}^{−1}}^{α_{1}^{−1}} \frac{t − α_{3}^{−1}}{s_{1}t^{2} − α_{1}^{−1}t} dt .

From (27), (79), (81), (82), (85), (86), (88) and (89) we get M_{i} ≥ 2e^{γ}𝔑_{i} {κ_{i} + O((log P)^{−1/4}))} , i = 1, 2 , (90) where

κ_{1} = s_{1}^{−1}( log(s_{1} − 1) − 2θ_{1} − 2θ_{1} ∫_{2}^{s_{1}−1} t^{−1} log(t − 1) dt ) − θ κ ,
κ_{2} = s_{2}^{−1}( log(s_{2} − 1) − θ_{2}) .

We choose

α_{1} = 0.004764 ,  α_{2} = 0.012346 ,  α_{3} = 0.014291 ,
θ = 0.12 ,  θ_{1} = 0.4405 ,  θ_{2} = 0.5595 ,
η_{1} = 0.019056 , η_{2} = 0.03395 .

It is easy to see that all conditions (10), (27), (36), (80) are satisfied. Moreover, using numerical integration we can verify that in this case

κ_{1} , κ_{2} > 0 . (91)

Using (84), (90) and (91) we find that M_{i} ≫ (log log P) (log P)^{−1} for i = 1, 2. Having in mind (76), (78), (84) and (85) we get M ≫ (log log P)^{3} (log P)^{−3}. Now we apply (4), (47) and (65) and the estimate (14) follows.

We have chosen our constants in such a way as to minimize the number max (α_{2}^{−1} , θ^{−1} + α_{3}^{−1}). It remains to note that our choice gives

80 < α_{2}^{−1} < 81 , 78 < θ^{−1} + α_{3}^{−1} < 79 .

We take into account (18), (19) and Theorem 1 is proved.

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Department of Mathematics Plovdiv University “P. Hilendarski” 24 “Tsar Asen” str. Plovdiv 4000 Bulgaria Email: dtolev@pu.acad.bg