ON LINEAR FORMS WITH COEFFICIENTS IN ℕ ζ(1 + ℕ)

The detailed version, Part 4

Leonid A. Gutnik

In memory of Professor Alexander Buchstab

Content

§0. Foreword. §1. For the left hand. §2 On one string. §3 A correction in [38].

§0. Foreword

Here I have collected the auxiliary results, which may be obtained by means of the elementary considerations described in [25].

§1. For the left hand.

For a prime number p, let v_{p} denotes the p-adic valuation on ℚ. Lemma 4.1.1. Let p is a prime number. Let

d ∈ ℕ − 1, r ∈ ℕ − 1, d_{1} ∈ ℕ − 1, d_{2} ∈ ℕ − 1, r_{1} ∈ ℕ − 1, r_{2} ∈ ℕ − 1,

and

(4.1.1)  max(r_{1}, r_{2}) < p.

Then

(4.1.2)  p^{−d}(dp + r)! ∈ (−1)^{d}d!r! + pℤ

and

(4.1.3)  \binom{(d_{1} + d_{2})p + r_{1} + r_{2}}{d_{1}p + r_{1}} ∈ \binom{d_{1} + d_{2}}{d_{1}}\binom{r_{1} + r_{2}}{r_{1}} + pℤ.

Typeset by AMS-TEX

Proof may be found in [25], Lemma 9.

Lemma 4.1.2. Let p is a prime number. Let

d ∈ ℕ, r ∈ ℕ, r < p, d^{∼} ∈ ℕ − 1,

and d^{∼} < d. Then

(4.1.4)  \binom{dp}{d^{∼}p + r} ∈ d\binom{d − 1}{d^{∼}}\binom{p}{r} + p^{2}ℤ.

Proof may be found in [25], Lemma 10.

Let l ∈ ℕ and p is arbitrary prime number in (5, +∞).

Since, according to (1.2.16), (i.e. Part 1, (2.16)),

(4.1.5)  α_{ν,k}^{∗} = (−1)^{ν+νδ+k}\binom{νδ}{k}^{3}\binom{νδ + k}{νδ − ν}^{3},

where k = 0 , …, νδ, it follows that, in view of the Lemma 4.1.1, if v_{p}(k) = 0, then

(4.1.6)  v_{p}(α_{pl,k}^{∗}) ≥ 3.

In view of the Lemma 4.1.1,

(4.1.7)  v_{p}(α_{pl,pd}^{∗} − α_{l,d}^{∗}) ≥ 1,

where d = 0 , …, lΔ, and therefore

(4.1.8)  v_{p}(α^{∗}(z, pl) − α^{∗}(z^{p}, l) ≥ 1.

If a ∈ ℤ, b ∈ ℕ, |a| ≥ b, z = a/b, p > |a| + b, then

(4.1.9)  v_{p}(z^{p}m − z^{m}) ≥ 1

for m ∈ ℤ and therefore

(4.1.10)  v_{p}(α^{∗}(z, pl) − α^{∗}(z, l) ≥ 1.

According to (1.2.17) and (1.2.18),

(4.1.11)  β_{ν,k}^{∗} = 3α_{ν,k}^{∗} (− ( ∑_{κ=ν+k+1}^{νΔ+k} 1/κ) − ( ∑_{κ=1}^{νΔ−k} 1/κ) + (∑_{κ=1}^{k} 1/κ)) ,

ON LINEAR FORMS WITH COEFFICIENTS IN ℕζ(1 + ℕ) 3

(4.1.12) 2γ_{ν,k}^{∗} = 9α_{ν,k}^{∗} (− ( ∑_{κ=ν+k+1}^{νΔ+k} 1/κ) − ( ∑_{κ=1}^{νΔ−k} 1/κ) + (∑_{κ=1}^{k} 1/κ))^{2} −

3α_{ν,k}^{∗} (( ∑_{κ=ν+k+1}^{νΔ+k} 1/κ^{2}) − ( ∑_{κ=1}^{νΔ−k} 1/κ^{2}) − (∑_{κ=1}^{k} 1/κ^{2})) ,

where k = 0 , …, νΔ. Let p > 2lΔ. In view of (4.1.6), (4.1.11) and (4.1.12), if k ∈ [0, plΔ] ∩ ℤ and v_{p}(k) = 0, then

(4.1.13)  v_{p}(β_{pl,k}^{∗}) ≥ 2, v_{p}(γ_{pl,k}^{∗}) ≥ 1.

Let 𝔒_{p} is the ring of all the p− integers of the field ℚ. Then

(4.1.14)  (− ( ∑_{κ=pl+k+1}^{plΔ+k} 1/κ) − ( ∑_{κ=1}^{plΔ−k} 1/κ) + (∑_{κ=1}^{k} 1/κ)) −

(− ( ∑_{\begin{matrix}κ=pl+k+1 \\ κ≡0 (mod p) \end{matrix}}^{plΔ+k} 1/κ) − ( ∑_{\begin{matrix}κ=1 \\ κ≡0 (mod p) \end{matrix}}^{plΔ−k} 1/κ) + ( ∑_{\begin{matrix}κ=1 \\ κ≡0 (mod p) \end{matrix}}^{k} 1/κ)) ∈ 𝔒_{p},

(4.1.15)  (− ( ∑_{κ=pl+k+1}^{plΔ+k} 1/κ) − ( ∑_{κ=1}^{plΔ−k} 1/κ) + (∑_{κ=1}^{k} 1/κ))^{2} −

(− ( ∑_{\begin{matrix}κ=pl+k+1 \\ κ≡0 (mod p) \end{matrix}}^{plΔ+k} 1/κ) − ( ∑_{\begin{matrix}κ=1 \\ κ≡0 (mod p) \end{matrix}}^{plΔ−k} 1/κ) + ( ∑_{\begin{matrix}κ=1 \\ κ≡0 (mod p) \end{matrix}}^{k} 1/κ))^{2} ∈ 𝔭^{−1}𝔒_{p},

(4.1.16)  (( ∑_{κ=pl+k+1}^{plΔ+k} 1/κ^{2}) − ( ∑_{κ=1}^{plΔ−k} 1/κ^{2}) − (∑_{κ=1}^{k} 1/κ^{2})) −

(( ∑_{\begin{matrix}κ=pl+k+1 \\ κ≡0 (mod p) \end{matrix}}^{plΔ+k} 1/κ^{2}) − ( ∑_{\begin{matrix}κ=1 \\ κ≡0 (mod p) \end{matrix}}^{plΔ−k} 1/κ^{2}) − ( ∑_{\begin{matrix}κ=1 \\ κ≡0 (mod p) \end{matrix}}^{k} 1/κ^{2})) ∈ 𝔒_{p},

and, if k = pd, with d = 0 , …, lΔ, then

(4.1.17) (− ( ∑_{\begin{matrix}κ=pl+pd+1 \\ κ≡0 (mod p) \end{matrix}}^{plΔ+pd} 1/κ) − ( ∑_{\begin{matrix}κ=1 \\ κ≡0 (mod p) \end{matrix}}^{plΔ−pd} 1/κ) + ( ∑_{\begin{matrix}κ=1 \\ κ≡0 (mod p) \end{matrix}}^{pd} 1/κ)) =

(− ( ∑_{δ=l+d+1}^{lΔ+d} 1/(pδ)) − ( ∑_{δ=1}^{lΔ−d} 1/(pδ)) + (∑_{κ=1}^{d} 1/(pδ))) =

p^{−1} (− ( ∑_{δ=l+d+1}^{lΔ+d} 1/δ) − ( ∑_{δ=1}^{lΔ−d} 1/δ) + (∑_{δ=1}^{d} 1/δ)) ,

(4.1.18) (− ( ∑_{\begin{matrix}κ=pl+pd+1 \\ κ≡0 (mod p) \end{matrix}}^{plΔ+pd} 1/κ) − ( ∑_{\begin{matrix}κ=1 \\ κ≡0 (mod p) \end{matrix}}^{plΔ−pd} 1/κ) + ( ∑_{\begin{matrix}κ=1 \\ κ≡0 (mod p) \end{matrix}}^{pd} 1/κ))^{2} =

p^{−2} (− ( ∑_{δ=l+d+1}^{lΔ+d} 1/δ) − ( ∑_{δ=1}^{lΔ−d} 1/δ) + (∑_{δ=1}^{d} 1/δ))^{2} ,

(4.1.19) (( ∑_{\begin{matrix}κ=pl+pd+1 \\ κ≡0 (mod p) \end{matrix}}^{plΔ+pd} 1/κ^{2}) − ( ∑_{\begin{matrix}κ=1 \\ κ^{2}≡0 (mod p) \end{matrix}}^{plΔ−pd} 1/κ) − ( ∑_{\begin{matrix}κ=1 \\ κ≡0 (mod p) \end{matrix}}^{pd} 1/κ^{2})) =

p^{−2} (( ∑_{δ=l+d+1}^{lΔ+d} 1/δ^{2}) − ( ∑_{δ=1}^{lΔ−d} 1/δ^{2}) − (∑_{δ=1}^{d} 1/δ^{2})) .

In view of (4.1.13) - (4.1.18), if p > 2lΔ, then

(4.1.20)  (− ( ∑_{κ=pl+k+1}^{plΔ+k} 1/κ) − ( ∑_{κ=1}^{plΔ−k} 1/κ) + (∑_{κ=1}^{k} 1/κ)) ∈ p^{−1}𝔒_{p},
(4.1.21)  (− ( ∑_{κ=pl+k+1}^{plΔ+k} 1/κ) − ( ∑_{κ=1}^{plΔ−k} 1/κ) + (∑_{κ=1}^{k} 1/κ))^{2} ∈ p^{−2}𝔒_{p},

ON LINEAR FORMS WITH COEFFICIENTS IN ℕζ(1 + ℕ) 5

(4.1.22)  (( ∑_{κ=pl+k+1}^{plΔ+k} 1/κ^{2}) − ( ∑_{κ=1}^{plΔ−k} 1/κ^{2}) + (∑_{κ=1}^{k} 1/κ^{2})) ∈ p^{−2}𝔒_{p}.

In view of (4.1.7),(4.1.11),(4.1.12), (4.1.14) - (4.1.22),

(4.1.23)  β_{pl,pd}^{∗} − p^{−1}β_{l,d}^{∗} =

3(α_{pl,pd}^{∗} − α_{l,d}^{∗}) (− ( ∑_{κ=pl+k+1}^{plΔ+k} 1/κ) − ( ∑_{κ=1}^{plΔ−k} 1/κ) + (∑_{κ=1}^{k} 1/κ)) +

3α_{l,d}^{∗}(( − ( ∑_{κ=pl+pd+1}^{plΔ+pd} 1/κ) − ( ∑_{κ=1}^{plΔ−pd} 1/κ) + ( ∑_{κ=1}^{pd} 1/κ) )−

p^{−1}( − ( ∑_{δ=l+d+1}^{lΔ+δ} 1/δ) − ( ∑_{δ=1}^{lΔ−d} 1/δ) + ( ∑_{κ=1}^{d} 1/δ))) ∈ 𝔒_{p},

(4.1.24)  2γ_{pl,pd}^{∗} − p^{−2}γ_{l,d}^{∗} =

9(α_{pl,pd}^{∗} − α_{l,d}^{∗}) (− ( ∑_{κ=pl+pd+1}^{plΔ+pd} 1/κ) − ( ∑_{κ=1}^{plΔ−pd} 1/κ) + (∑_{κ=1}^{pd} 1/κ))^{2} −

3(α_{pl,pd}^{∗} − α_{l,d}^{∗}) (( ∑_{κ=pl+pd+1}^{plΔ+pd} 1/κ^{2}) − ( ∑_{κ=1}^{plΔ−pd} 1/κ^{2}) − (∑_{κ=1}^{pd} 1/κ^{2})) +

9α_{l,d}^{∗}(( − ( ∑_{κ=pl+pd+1}^{plΔ+pd} 1/κ) − ( ∑_{κ=1}^{plΔ−pd} 1/κ) + ( ∑_{κ=1}^{pd} 1/κ) )^{2}−

p^{−2} (− ( ∑_{δ=l+d+1}^{lΔ+δ} 1/δ) − ( ∑_{δ=1}^{lΔ−d} 1/δ) + ( ∑_{κ=1}^{d} 1/δ))^{2} )−

3α_{l,d}^{∗}((( ∑_{κ=pl+pd+1}^{plΔ+pd} 1/κ^{2}) − ( ∑_{κ=1}^{plΔ−pd} 1/κ^{2}) + ( ∑_{κ=1}^{pd} 1/κ^{2}))−

p^{−2}( − ( ∑_{δ=l+d+1}^{lΔ+δ} 1/δ^{2}) − ( ∑_{δ=1}^{lΔ−d} 1/δ^{2}) − ( ∑_{δ=1}^{d} 1/δ^{2}))) ∈ p^{−1}𝔒_{p}.

6 LEONID A. GUTNIK

In view of (4.1.13), (4.1.23), (4.1.24),

(4.1.25) v_{p}(β^{∗}(z; pl) − p^{−1}β^{∗}(z^{p}; l)) ≥ 0, v_{p}(γ^{∗}(z, pl) − p^{−2}γ^{∗}(z^{p}, l)) ≥ −1

and, if a ∈ ℤ, b ∈ ℕ, |a| > 0, z = a/b, p > 2lΔ + |a| + b, in view of (4.1.9),

(4.1.26)  v_{p}(β^{∗}(z; pl) − p^{−1}β^{∗}(z; l)) ≥ 0, v_{p}(γ^{∗}(z, pl) − p^{−2}γ^{∗}(z, l)) ≥ −1.

According to (1.2.24), (1.2.26) and (1.2.28),

(4.1.27)  ϕ^{∗}(z; ν) = (−z)^{ν} ∑_{k=0}^{νΔ} α_{ν,k}^{∗}z^{k} ∑_{t=1}^{k+ν} z^{−t}t^{−3} +

+ (−z)^{ν} ∑_{k=0}^{νΔ} β_{ν,k}^{∗}z^{k} ∑_{t=1}^{k+ν} z^{−t}t^{−2} + (−z)^{ν} ∑_{k=0}^{νΔ} γ_{ν,k}^{∗}z^{k} ∑_{t=1}^{k+ν} z^{−t}t^{−1} =

= (−z)^{ν} ∑_{t=1}^{ν} z^{−t}(t^{−3}α^{∗}(z; ν) + t^{−2}β^{∗}(z; ν) + t^{−1}γ^{∗}(z; ν)) +

+ (−z)^{ν} ∑_{k=0}^{νΔ} α_{ν,k}^{∗}z^{k} ∑_{t=1+ν}^{k+ν} z^{−t}t^{−3} + (−z)^{ν} ∑_{k=0}^{νΔ} β_{ν,k}^{∗}z^{k} ∑_{t=1+ν}^{k+ν} z^{−t}t^{−2} +

+ (−z)^{ν} ∑_{k=0}^{νΔ} γ_{ν,k}^{∗}z^{k} ∑_{t=1+ν}^{k+ν} z^{−t}t^{−1} =

= (−z)^{ν} ∑_{t=1}^{ν} z^{−t}(t^{−3}α^{∗}(z; ν) + t^{−2}β^{∗}(z; ν) + t^{−1}γ^{∗}(z; ν)) +

(−z)^{ν} ∑_{t=1+ν}^{νΔ+ν} z^{−t}t^{−3} ∑_{k=t−ν}^{νΔ} α_{ν,k}^{∗}z^{k} + (−z)^{ν} ∑_{t=1+ν}^{νΔ+ν} z^{−t}t^{−2} ∑_{k=t−ν}^{νΔ} β_{ν,k}^{∗}z^{k}+

(−z)^{ν} ∑_{t=1+ν}^{νΔ+ν} z^{−t}t^{−1} ∑_{k=t−ν}^{νΔ} γ_{ν,k}^{∗}z^{k};

(4.1.28)  ψ^{∗}(z; ν) = (−z)^{ν} ∑_{k=0}^{νΔ} 3α_{ν,k}^{∗} z^{k} ∑_{t=1}^{k+ν} z^{−t}t^{−4} +

+ (−z)^{ν} ∑_{k=0}^{νΔ} 2β_{ν,k}^{∗} z^{k} ∑_{t=1}^{k+ν} z^{−t}t^{−3} + (−z)^{ν} ∑_{k=0}^{νΔ} γ_{ν,k}^{∗} z^{k} ∑_{t=1}^{k+ν} z^{−t}t^{−2} =

ON LINEAR FORMS WITH COEFFICIENTS IN ℕζ(1 + ℕ) 7

= (−z)^{ν} ∑_{t=1}^{ν} z^{−t}(t^{−4}3α^{∗}(z; ν) + t^{−3}2β^{∗}(z; ν) + t^{−2}γ^{∗}(z; ν)) +

+ (−z)^{ν} ∑_{k=0}^{νΔ} 3α_{ν,k}^{∗}z^{k} ∑_{t=1+ν}^{k+ν} z^{−t}t^{−4} + (−z)^{ν} ∑_{k=0}^{νΔ} 2β_{ν,k}^{∗}z^{k} ∑_{t=1+ν}^{k+ν} z^{−t}t^{−3} +

+ (−z)^{ν} ∑_{k=0}^{νΔ} γ_{ν,k}^{∗}z^{k} ∑_{t=1+ν}^{k+ν} z^{−t}t^{−2} =

= (−z)^{ν} ∑_{t=1}^{ν} z^{−t}(t^{−4}3α^{∗}(z; ν) + t^{−3}2β^{∗}(z; ν) + t^{−2}γ^{∗}(z; ν)) +

(−z)^{ν} ∑_{t=1+ν}^{νΔ+ν} z^{−t}t^{−4} ∑_{k=t−ν}^{νΔ} 3α_{ν,k}^{∗}z^{k} + (−z)^{ν} ∑_{t=1+ν}^{νΔ+ν} z^{−t}t^{−3} ∑_{k=t−ν}^{νΔ} 2β_{ν,k}^{∗}z^{k}+

(−z)^{ν} ∑_{t=1+ν}^{νΔ+ν} z^{−t}t^{−2} ∑_{k=t−ν}^{νΔ} γ_{ν,k}^{∗}z^{k};

(4.1.29)  ξ^{∗}(z; ν) = (−z)^{ν} ∑_{k=0}^{νΔ} 6α_{ν,k}^{∗}z^{k} ∑_{t=1}^{k+ν} z^{−t}t^{−5} +

+ (−z)^{ν} ∑_{k=0}^{νΔ} 3β_{ν,k}^{∗}z^{k} ∑_{t=1}^{k+ν} z^{−t}t^{−4} + (−z)^{ν} ∑_{k=0}^{νΔ} γ_{ν,k}^{∗}z^{k} ∑_{t=1}^{k+ν} z^{−t}t^{−3} =

= (−z)^{ν} ∑_{t=1}^{ν} z^{−t}(6t^{−5}α^{∗}(z; ν) + 3t^{−4}β^{∗}(z; ν) + t^{−3}γ^{∗}(z; ν)) +

+ (−z)^{ν} ∑_{k=0}^{νΔ} 6α_{ν,k}^{∗}z^{k} ∑_{t=1+ν}^{k+ν} z^{−t}t^{−5} + (−z)^{ν} ∑_{k=0}^{νΔ} 3β_{ν,k}^{∗}z^{k} ∑_{t=1+ν}^{k+ν} z^{−t}t^{−4} +

+ (−z)^{ν} ∑_{k=0}^{νΔ} γ_{ν,k}^{∗}z^{k} ∑_{t=1+ν}^{k+ν} z^{−t}t^{−3} =

= (−z)^{ν} ∑_{t=1}^{ν} z^{−t}(t^{−5}6α^{∗}(z; ν) + t^{−4}3β^{∗}(z; ν) + t^{−3}γ^{∗}(z; ν)) +

(−z)^{ν} ∑_{t=1+ν}^{νΔ+ν} z^{−t}t^{−5} ∑_{k=t−ν}^{νΔ} 6α_{ν,k}^{∗}z^{k} + (−z)^{ν} ∑_{t=1+ν}^{νΔ+ν} z^{−t}t^{−4} ∑_{k=t−ν}^{νΔ} 3β_{ν,k}^{∗}z^{k}+

(−z)^{ν} ∑_{t=1+ν}^{νΔ+ν} z^{−t}t^{−3} ∑_{k=t−ν}^{νΔ} γ_{ν,k}^{∗}z^{k}.

According to (4.1.10), (4.1.25) - (4.1.29), if a ∈ ℤ, b ∈ ℕ, |a| > 0, z = a/b, p > 2lΔ + |a| + b, then

(4.1.30)  v_{p}(ϕ^{∗}(z; pl) − p^{−3}ϕ^{∗}(z; l)) ≥ −2,

(4.1.31)  v_{p}(ψ^{∗}(z; pl) − p^{−4}ψ^{∗}(z; l)) ≥ −3,

(4.1.32)  v_{p}(ξ^{∗}(z; pl) − p^{−5}ξ^{∗}(z; l)) ≥ −4.

Let

α_{0}^{∗}(z; ν) = α^{∗}(z; ν), α_{1}^{∗}(z; ν) = β^{∗}(z; ν), α_{2}^{∗}(z; ν) = γ^{∗}(z; ν),

(4.1.33)

α_{3}^{∗}(z; ν) = ϕ^{∗}(z; ν), α_{4}^{∗}(z; ν) = ψ^{∗}(z; ν), α_{5}^{∗}(z; ν) = ξ^{∗}(z; ν),

where ν ∈ ℕ. Then (4.1.10), (4.1.26), (4.1.30) - (4.1.32) may be rewritten in the form

(4.1.33)  v_{p}(α_{j}^{∗}(z; pl) − p^{−j}α_{j}^{∗}(z; l)) ≥ 1 − j,

where j = 0 , …, 5.

Lemma 4.1.3. Let k_{j} ∈ ℕ − 1, where j = 0 , …, 5, ∑ j = 1^{5} > 0, J = {j ∈ {0 , …, 5}: k_{j} > 0} and ∏_{j∈J} α_{j}^{∗}(z; l_{j}) ≠ 0 for some z ∈ ℚ⧹{0} and l_{j}) ∈ ℕ, with j ∈ J. Then, for each m ∈ ℕ the sequences

(4.1.34)  α_{j}^{∗}(z; m + 1) , …, α_{j}^{∗}(z; m + μ) , …

with j ∈ J compose a linearly independent system over ℂ.

Proof. Since z ∈ ℚ, it follows that α_{j}^{∗}(z; l_{j}) ∈ ℚ for j ∈ J and there exists d ∈ ℕ, such that dα_{j}^{∗}(z; l_{j}) ∈ ℕ for j ∈ J. Since α_{j}^{∗}(z; ν) ∈ ℚ for j ∈ J and ν ∈ ℕ, it follows that it is sufficient to prove the linear independence of the system (4.1.34) over ℚ. The contrary means the existence b_{j} ∈ ℤ, where j ∈ J, such that

(4.1.35)  ∑_{j∈J} |b_{j}| > 0, ∑_{j∈J} b_{j}α_{j}^{∗}(z; ν) = 0,

where ν ∈ m + ℕ. Let k = sup{j ∈ J: b_{j} ≠ 0} and let p is the prime number such that

p > ∑_{j∈J} d|b_{j}α_{j}^{∗}(z; ν)| + m + 1 + d.

We take now in (4.1.33) ν = pl_{k}. Then according (4.1.35), (4.1.33),

−k = v_{p}(α_{j}^{∗}(z; pl_{k})) = v_{p} ( ∑_{j∈J⧹{k}} b_{j}α_{j}^{∗}(z; ν)) ≥ 1 − k.

■

Lemma 4.1.4. Let Δ is equal to the prime number p > 5. If z = 1, then condition of the Lemma 4.1.3 is fulfilled for J = {0, 1, 2, 4, 5}. If z = −1, then condition of the Lemma 4.1.3 is fulfilled for J = {0, 1, 2, 3, 4, 5}.

Proof. In view of (4.1.5),

(4.1.36)  α_{1,0}^{∗} = p^{3}.

and,for k = 1 , …, p there exist x_{k} ∈ 𝔒_{p}, such that

(4.1.37)  α_{1,p−1}^{∗} = p^{3} (∏_{k=1}^{p−1}(1 + \frac{p}{k} )^{3} ∈ p^{3} + p^{5}𝔒_{p},

(4.1.38)  α_{1,p}^{∗} = −8p^{3} (∏_{k=1}^{p−1}(1 + \frac{p}{k} )^{3} /(1 + p)^{3} ∈ −8p^{3} + 24p^{4} + p^{5}𝔒_{p},

(4.1.39)  α_{1,k}^{∗} ∈ p^{6}𝔒_{p},

where k = 1 , …, p − 2. According to (1.2.21),

(4.1.40) α^{∗}(z; ν) = (−z)^{ν} ∑_{k=0}^{νΔ} α_{ν,k}^{∗}z^{k}, β^{∗}(z; ν) = (−z)^{ν} ∑_{k=0}^{νΔ} β_{ν,k}^{∗}z^{k},

γ^{∗}(z; ν) = (−z)^{ν} ∑_{k=0}^{νΔ} γ_{ν,k}^{∗}z^{k},

where ν ∈ ℕ − 1. In view of (4.1.40), (4.1.36) - (4.1.39),

(4.1.41)  α^{∗}(1; 1) ∈ 6p^{3} − 24p^{4} + p^{5}𝔒_{p}, α^{∗}(−1; 1) ∈ 10p^{3} − 24p^{4} + p^{5}𝔒_{p}.

10 LEONID A. GUTNIK

In view of (4.1.11), (4.1.12), (4.1.36) - (4.1.39),

(4.1.42)  β_{1,0}^{∗} = 3α_{1,0}^{∗} (− (∑_{κ=2}^{p} 1/κ) − (∑_{κ=1}^{p} 1/κ)) ∈

3p^{3}(−\frac{2}{p} + 1 + p𝔒_{p}) = 3p^{2}(−2 + p + p^{2}𝔒_{p},

− ( ∑_{κ=p+1}^{2p−1} 1/κ) − 1 + (∑_{κ=1}^{p−1} 1/κ) =

−1 + p ∑_{κ=1}^{p−1}(\frac{1}{κ(p + κ)} ∈ −1 − p^{2}𝔒_{p},

(4.1.43)  β_{1,p−1}^{∗} = 3α_{1,p−1}^{∗} (− ( ∑_{κ=p+1}^{2p−1} 1/κ) − 1 + (∑_{κ=1}^{p−1} 1/κ)) ∈

−3p^{3}(1 + p^{2}𝔒_{p})(1 + p^{2}𝔒_{p}) = −3p^{3}(1 + p^{2}𝔒_{p}),

− ( ∑_{κ=p+2}^{2p} 1/κ) + (∑_{κ=1}^{p} 1/κ) =

\frac{1}{2p} + \frac{1}{p + 1} + p (∑_{κ=1}^{p−1} \frac{1}{κ(p + κ)}) ∈ \frac{1}{2p} + 1 − p + p^{2}𝔒_{p} = \frac{1}{2p}(1 + 2p − 2p^{2} + p^{3}𝔒_{p}),

(4.1.44)  β_{1,p}^{∗} = 3α_{1,p}^{∗} (− ( ∑_{κ=p+2}^{2p} 1/κ) + (∑_{κ=1}^{p} 1/κ)) ∈

−12p^{2}(1 − 3p + p^{2}𝔒_{p})(1 + 2p − p^{2} + p^{3}𝔒_{p})) = −12p^{2}(1 − p + p^{2}𝔒_{p}),

(4.1.45) β_{1,k}^{∗} = 3α_{1,k}^{∗} (− ( ∑_{κ=k+2}^{p+k} 1/κ) − (∑_{κ=1}^{p−k} 1/κ) + (∑_{κ=1}^{k} 1/κ)) ∈ p^{5}𝔒_{p},

where k = 1 , …, p − 2,

(4.1.46) 2γ_{1,0}^{∗} = 9α_{1,0}^{∗}( − ( ∑_{κ=2}^{p} 1/κ) − ( ∑_{κ=1}^{p} 1/κ))^{2}−

3α_{1,0}^{∗}( ∑_{κ=2}^{p} 1/κ^{2}) − ( ∑_{κ=1}^{p} 1/κ^{2})) ∈

ON LINEAR FORMS WITH COEFFICIENTS IN ℕζ(1 + ℕ) 11

9p(−2 + p + p^{2}𝔒_{p})^{2} − 3p^{3} ⊂ 36p(1 − p + p^{2}𝔒_{p}),

( ∑_{κ=p+1}^{2p−1} 1/κ^{2}) − (∑_{κ=1}^{p−1} 1/κ^{2}) =

−p (∑_{κ=1}^{p−1} \frac{2κ + p}{κ^{2}(κ + p)^{2}} ) ∈ p^{2}𝔒_{p},

(4.1.47) 2γ_{1,p−1}^{∗} = 9α_{1,p−1}^{∗} (− ( ∑_{κ=p+1}^{2p−1} 1/κ) − 1 + (∑_{κ=1}^{p−1} 1/κ))^{2} −

3α_{1,p−1}^{∗} (( ∑_{κ=p+1}^{2p−1} 1/κ^{2}) − 1 − (∑_{κ=1}^{p−1} 1/κ^{2})) ∈

9p^{3}(1 + p^{2}𝔒_{p})(1 + p^{2}𝔒_{p})^{2} − 3p^{3}(1 + p^{2}𝔒_{p})(−1 + p^{2}𝔒_{p}) = 12p^{3}(1 + p^{2}𝔒_{p}),

( ∑_{κ=p+2}^{2p} 1/κ^{2}) − (∑_{κ=1}^{p} 1/κ^{2}) =

−\frac{3}{4p^{2}} − \frac{1}{(p + 1)^{2}} − p (∑_{κ=1}^{p−1} \frac{p + 2κ}{κ^{2}(p + κ)^{2}} ) =

−\frac{3}{4p^{2}} − 1 + 2p − p^{2}x_{2p+8} ∈ −\frac{1}{4p^{2}} (3 + 4p^{2} − 8p^{3} + 4p^{4}𝔒_{p}),

(4.1.48) 2γ_{1,p}^{∗} = 9α_{1,p}^{∗} (− ( ∑_{κ=p+2}^{2p} 1/κ) + (∑_{κ=1}^{p} 1/κ))^{2} −

3α_{1,p}^{∗} (( ∑_{κ=p+2}^{2p} 1/κ^{2}) − (∑_{κ=1}^{p} 1/κ^{2})) ∈

− 18p(1 − 3p + p^{2}𝔒_{p})(1 + 2p − 2p^{2} + p^{3}𝔒_{p})^{2}− 6p(1 − 3p + p^{2}x_{p})(3 + 4p^{2} − 8p^{3} + 4p^{4}x_{2p+8}) = p(−36 + 36p + p^{2}𝔒_{p}),

(4.1.49)  2γ_{1,k}^{∗} = 9α_{1,k}^{∗} (− ( ∑_{κ=2+k}^{p+k} 1/κ) − (∑_{κ=1}^{p−k} 1/κ) + (∑_{κ=1}^{k} 1/κ))^{2} −

3α_{ν,k}^{∗} (( ∑_{κ=2+k}^{p+k} 1/κ^{2}) − (∑_{κ=1}^{p−k} 1/κ^{2}) − (∑_{κ=1}^{k} 1/κ^{2})) ∈ p^{4}𝔒_{p},

where k = 1 , …, p − 2.

In view of (4.1.40), (4.1.42) - (4.1.45),

(4.1.50)  β^{∗}(1, 1) ∈ 18p^{2} − 12p^{3} + p^{4}𝔒_{p},

(4.1.51)  β^{∗}(−1, 1) ∈ 6p^{2} − 12p^{3} + p^{4}𝔒_{p}.

In view of (4.1.46) - (4.1.49),

(4.1.52)  2γ^{∗}(ε, 1) ∈ 36(1 − ε)p(1 − p) + p^{3}𝔒_{p},

where ε^{2} = 1. In view of (4.1.27) - (4.1.29), (4.1.36) - (4.1.39), (4.1.41), (4.1.50) - (4.1.52), if ε^{2} = 1, then

ϕ^{∗}(ε; 1) = −(α^{∗}(ε; 1)β^{∗}(ε; 1) + γ^{∗}(ε; 1)) +

(−ε) ∑_{t=2}^{p+1} ε^{−t}t^{−3} ∑_{k=t−1}^{p} α_{1,k}^{∗}ε^{k} + (−ε) ∑_{t=1+2}^{p+1} ε^{−t}t^{−2} ∑_{k=t−1}^{p} β_{1,k}^{∗}ε^{k}+

(−ε) ∑_{t=2}^{p+1} ε^{−t}t^{−1} ∑_{k=t−1}^{p} γ_{1,k}^{∗}ε^{k} ∈

−p^{−3}(α_{1,p−1}^{∗} + α_{1,p}^{∗}ε) − p^{−2}(β_{1,p−1}^{∗} + β_{1,p}^{∗}ε) − p^{−1}(γ_{1,p−1}^{∗} + γ_{1,p}^{∗}ε) + p𝔒_{p} ⊂

−1 + 38ε + p𝔒_{p},
ψ^{∗}(ε; 1) ∈

−p^{−4}(3α_{1,p−1}^{∗} + 3α_{1,p}^{∗}ε) − p^{−3}(2β_{1,p−1}^{∗} + 2β_{1,p}^{∗}ε) − p^{−2}(γ_{1,p−1}^{∗} + γ_{1,p}^{∗}ε) + p𝔒_{p} ⊂

p^{−1}(−3 + 66ε) + 𝔒_{p},
ξ^{∗}(ε; 1) ∈

−p^{−5}(6α_{1,p−1}^{∗} + 6α_{1,p}^{∗}ε) − p^{−4}(3β_{1,p−1}^{∗} + 3β_{1,p}^{∗}ε) − p^{−3}(γ_{1,p−1}^{∗} + γ_{1,p}^{∗}ε) + p𝔒_{p} ⊂

p^{−2}(−6 + 102ε) + p^{−1}𝔒_{p}.

■

According to the Lemmata 4.1.3 and 4.1.4, for each m ∈ ℕ the sequences of numbers

(4.1.53)  α_{j}^{∗}(−1; m + 1), …, α_{j}^{∗}(−1; m + μ), …

where j = 0, 1, 2, 3, 4, 5, compose a linear independent system over ℂ. Consequently, for each m ∈ ℕ the sequences of polynomials

(4.1.54)  α_{j}^{∗}(z; m + 1) , …, α_{j}^{∗}(z; m + μ) , …

with j = 0, 1, 2, 3, 4, 5 compose a linear independent system over ℂ(z). Therefore for the equation (3.1.106) we have the inequality

(4.1.55)  q_{6}(z; ν^{−1})q_{0}(z; ν^{−1}) ∈ ℂ[z]⧹0.

§2. On one string.

Lemma 4.2.1. If z ≥ 1, then

lim_{ν→∞}(|f_{2}^{∗}(z, ν)|)^{1/ν} = −h^{∼}(η_{1}^{∧}(z, π, δ_{0})),

lim_{ν→∞} f_{2+2k}^{∗}(z, ν)/f_{2}^{∗}(z, ν) = 2((logz)/2)^{k},

where k = 1, 2, and h^{∼}(η) is defined in (2.4.55).

Proof Together with the function

(4.2.1)  R_{0}(t, ν) = \frac{∏_{κ=1+ν}^{νΔ}(t − κ)}{∏_{κ=0}^{νΔ}(t + κ)} \frac{(νΔ)!}{(νd_{1})!}

introduced in (1.2.6) we consider also the rational functions

R_{1}(t, ν) = R_{0}(t, ν)(t + νΔ), R_{2}(t, ν) = (t + νΔ)^{−1},

where ν ∈ ℕ − 1. Then

(4.2.2)  R_{0}(t, ν) = R_{1}(t, ν)R_{2}(t, ν),

where ν ∈ ℕ − 1. If z ≥ 1, then, in view of (1.2.6), (1.2.15)

(4.2.3)  |f_{2}^{∗}(z, ν)| = z^{ν} ∑_{t=1+νΔ}^{+∞} z^{−t}R_{0}(t; ν)^{3},

(4.2.4)  |f_{2}^{∗}(z, ν)| ≤ z^{ν} ( ∑_{t=1+νΔ}^{+∞} R_{2}(t; ν)^{3}) ×

sup{z^{−t}R_{1}(t; ν)^{3}: t ∈ ℤ, t ≥ 1 + νΔ} ≤
z^{ν} (∫_{2νΔ}^{+∞} z^{−t}t^{−3}) ×
sup{z^{−t}R_{1}(t; ν)^{3}: t ∈ ℤ, t ≥ 1 + νΔ} =
z^{ν}2^{−3}(νΔ)^{−2} sup{z^{−t}R_{1}(t; ν)^{3}: t ∈ ℤ, t ≥ 1 + νΔ}.

Let t = ντΔ. Since

(4.2.5) z^{−t−1}R_{1}(t + 1; ν)^{3} = z^{−t}R_{1}(t; ν)^{3}z^{−1}t^{3}(t − ν)^{3}(t − νΔ)^{−3}(t + νΔ)^{−3},

it follows that

(4.2.6) z^{−t−1}R_{1}(t + 1; ν)^{3} = z^{−t}R_{1}(t; ν)^{3}z^{−1}τ^{3}(τ − δ_{0})^{3}(τ − 1)^{−3}(τ + 1)^{−3},

where, as in Part 1, paragraph 2, δ_{0} = 1/Δ. Let

(4.2.7)  g(z, τ) = z^{−1}τ^{3}(τ − δ_{0})^{3}(τ − 1)^{−3}(τ + 1)^{−3} − 1,

where 1 ≤ z, 1 ≤ τ. The substitution

(4.2.8)  τ = (1 + η)/(1 − η)

transforms the function (4.2.7) into the function

(4.2.9)  h(z, η) = g(z, (1 + η)/(1 − η)) =
(1 + δ_{0})^{3}2^{−6}z^{−1}((η + 1)^{3}(η + γ_{1})^{3} − 2^{3}(1 + γ_{1})^{3}η^{3}z) =
(1 + δ_{0})^{3}2^{−6}z^{−1}D^{∧}(z, η),

where, as in Part 1, paragraph 2, γ_{1} = (1 − δ_{0})/(1 + δ_{0}) and D^{∧}(z, η), as in (3.0.1). Clearly, for fixed z = |z| = r^{3} ≥ 1, the polynomial D^{∧}(z, η), has only two real roots η_{k}^{∧}(r, π, δ_{0}, where k = 0, 1, and, since we consider now δ_{0} ∈ (0, 1), according to (3.1.55), it follows that

(4.2.10)  0 < η_{1}^{∧}(r, π, δ_{0}) = γ_{1}/η_{0}(r, π, δ_{0} ≤ γ_{1} < 1 ≤ η_{0}(r, π, δ_{0},

h(z, η) > 0, if η ∈ (0, η_{1}^{∧}(r, π, δ_{0}), and h(z, η) < 0, if η ∈ (η_{1}^{∧}(r, π, δ_{0}), 1). Therefore the defined in (4.2.7) function g(z, τ) has onto (0, +∞) the single root

(4.2.11)  τ_{1} = τ_{1}(r, δ_{0}) = (1 + η_{1}^{∧}(r, π, δ_{0}))/(1 − η_{1}^{∧}(r, π, δ_{0})),

where r = z^{1/3} ≥ 1, δ_{0} ∈ (0, 1); clearly δ_{0} < 1 < τ_{1}; moreover g(z, τ) > 0, if τ ∈ (0, τ_{1}(r, δ_{0})) and g(z, τ) < 0, if τ ∈ (τ_{1}(r, δ_{0}), +∞). Consequently,

(R_{1}(t + 1, ν))^{3}z^{−t−1} ≤ (R_{1}(t, ν))^{3}z^{−t}

for t ≥ νΔτ_{1} > ν and

R_{1}(t + 1, ν)|z|^{−t−1} ≥ R_{1}(t, ν)|z|^{−t}

for νΔ + 1 ≤ t ≤ νΔτ_{1}; these inequalities are fulfilled for νΔ(τ_{1} − 1) ≥ 1, i.e

(4.2.12)  ν ≥ (Δ(τ_{1} − 1))^{−1},

and we consider further only such large ν. Therefore

(4.2.13) sup{z^{−t}R_{1}(t; ν)^{3}: t ∈ ℤ, t ≥ 1 + νΔ} = z^{−νΔτ_{1}−θ_{ν}} R_{1}(νΔτ_{1} + θ_{ν}; ν)^{3},

where νΔτ_{1} + θ_{ν} ∈ ℕ, O ≤ θ_{ν} < 1. Since all the summands of the sum (4.2.3) are not less than 0, in view of (4.2.4) and (4.2.13),it follows that

(4.2.14)  z^{ν}z^{−νΔτ_{1}−θ_{ν}} (νΔτ_{1} + θ_{ν} + νΔ)^{−3}(R_{1}(νΔτ_{1} + θ_{ν}; ν))^{3} =
z^{ν}z^{[−νΔτ_{1}]}(−[−νΔτ_{1}] + νΔ)^{−3}R_{1}(−[−νΔτ_{1}]; ν)^{3} ≤

z^{ν} ∑_{t=1+νΔ}^{+∞} z^{−t}R_{0}(t; ν)^{3} = |f_{2}^{∗}(z, ν)| ≤

z^{ν}2^{−3}(νΔ)^{−2}z^{−νΔτ_{1}−θ_{ν}} (R_{1}(νΔτ_{1} + θ_{ν}; ν))^{3}

In order to study the value

z^{−t}(R_{1}(t))^{3} = z^{−t} (\frac{Γ(t + Δν)Γ(t − ν)}{Γ(t − Δν)Γ(t)} \frac{(νΔ)!}{(νd_{1})!})^{3}

we use the Stirling’s formula ([19], chapter IV, 4.4.2). Then we see that

(4.2.15)  log(z^{−t}(R_{1}(t; ν))^{3}) = 3((ντΔ − 1/2) log(ντΔ) − ντΔ)−
3((ν(τ − 1)Δ − 1/2) log(ν(τ − 1)Δ) − ν(τ − 1)Δ)−

16 LEONID A. GUTNIK

3((ν(τ + 1)Δ − 1/2) log(ν(τ + 1)Δ) − ν(τ + 1)Δ)+

3((ν(τ − δ_{0})Δ − 1/2) log(ν(τ − δ_{0})Δ) − ν(τ − δ_{0})Δ) + 3((νΔ + 1/2) log(νΔ) − νΔ)− 3((ν(1−δ_{0})Δ−1/2) log(ν(1−δ_{0})Δ)−ν(1−δ_{0})Δ)−(ντΔ) log(z)+3θ_{1}^{∗}(Δ, τ, ν)/4 =

(ντΔ − \frac{1}{2} − 2 (ντΔ − \frac{1}{2}) + ν(τ − δ_{0})Δ − \frac{1}{2} + νΔ + \frac{1}{2} − (ν(1 − δ_{0})Δ − \frac{1}{2}) ×

3(log(νΔ)) + 3(ντΔ − 1/2) log(τ) − 3((ν(τ − 1)Δ − 1/2) log(τ − 1)−
3((ν(τ + 1)Δ − 1/2) log(τ + 1) + 3((ν(τ − δ_{0})Δ − 1/2) log(τ − δ_{0})−
ν(−δ_{0})Δ) + 3(−νΔ)−

3((ν(1 − δ_{0})Δ − 1/2) log(1 − δ_{0}) − ν(1 − δ_{0})Δ) − (ντΔ) log(z) + 3θ_{1}^{∗}(Δ, τ, ν)/4 =

3ντΔ log(τ) − 3ν(τ − 1)Δ log(τ − 1)−

3ν(τ + 1)Δ log(τ + 1) + 3(ν(τ − δ_{0})Δ log(τ − δ_{0})−
3(ν(1 − δ_{0})Δ log(1 − δ_{0})) − (ντΔ) log(z)−
(3/2) log((τ^{2} − τδ_{0})(1 − δ_{0})/(τ^{2} − 1)) + 3θ_{1}^{∗}(Δ, τ, ν)/4,

where ν(τ − 1)Δ ≥ 1 and |θ_{1}^{∗}(Δ, τ, ν)| ≤ 1. Let

(4.2.16)  u(z, τ) = 3τΔ log(τ) − 3(τ − 1)Δ log(τ − 1)−
3(τ + 1)Δ log(τ + 1) + 3(τ − δ_{0})Δ log(τ − δ_{0})−
3((1 − δ_{0})Δ log(1 − δ_{0})) − τΔ log(z),

where τ > 1 and z ≥ 1. In view of (4.2.15) and (4.2.16)

(4.2.17)  log(z^{−ντΔ}(R_{1}(ντΔ; ν))^{3}) = u(z, τ)ν−
(3/2) log((τ^{2} − τδ_{0})(1 − δ_{0})/(τ^{2} − 1)) + 3θ_{1}^{∗}(Δ, τ, ν)/4,

where ν(τ − 1)Δ ≥ 1 and |θ_{1}^{∗}(Δ, τ, ν)| ≤ 1. Let

(4.2.18)  w_{1}(Δ, z) = u(z, τ_{1}) = 3τ_{1}Δ log(τ_{1}) − 3(τ_{1} − 1)Δ log(τ_{1} − 1)−
3(τ_{1} + 1)Δ log(τ_{1} + 1) + 3(τ_{1} − δ_{0})Δ log(τ_{1} − δ_{0})−
3((1 − δ_{0})Δ log(1 − δ_{0})) − τ_{1}Δ log(z),

where z ≥ 1. If t = ντΔ, τ = τ_{1} + θ_{ν}/(νΔ) then

(4.2.19)  log(z^{−τ_{1}+θ_{ν}/(νΔ)}(R_{1}(τ_{1} + θ_{ν}/(νΔ)))^{3}) =

ON LINEAR FORMS WITH COEFFICIENTS IN ℕζ(1 + ℕ) 17

3ν(τ_{1} + θ_{ν}/(νΔ))Δ log(τ_{1} + θ_{ν}/(νΔ))− 3ν(τ_{1} + θ_{ν}/(νΔ) − 1)Δ log(τ_{1} + θ_{ν}/(νΔ) − 1)− 3ν(τ_{1} + θ_{ν}/(νΔ) + 1)Δ log(τ_{1} + θ_{ν}/(νΔ) + 1)+ 3(ν(τ_{1} + θ_{ν}/(νΔ) − δ_{0})Δ log(τ_{1} + θ_{ν}/(νΔ) − δ_{0})− 3(ν(1 − δ_{0})Δ log(1 − δ_{0})) − (ν(τ_{1} + θ_{ν}/(νΔ))Δ) log(z)− −(3/2) log((τ^{2} − τδ_{0})(1 − δ_{0})/(τ^{2} − 1)) + 3θ_{1}^{∗}(Δ, τ, ν)/4 =

3ν(τ_{1})Δ log(τ_{1}) − 3ν(τ_{1} − 1)Δ log(τ_{1} − 1)−

3ν(τ_{1} + 1)Δ log(τ_{1} + 1) + 3(ν(τ_{1} − δ_{0})Δ log(τ_{1} − δ_{0})−

3(ν(1 − δ_{0})Δ log(1 − δ_{0})) − (ντ_{1}Δ) log(z) + O(1) = w_{1}(Δ, z)ν + O(1),

when ν → ∞. Since g(z, τ_{1}) = 0, it follows that, according to (4.2.7),

(4.2.20) − log(z) + 3 log(τ_{1}) + 3 log(τ_{1} − δ_{0}) − 3 log(τ_{1} − 1) − 3 log(τ_{1} + 1) = 0.

Multiplying (4.2.20) by the number Δ(τ_{1} − δ_{0}) and subtracting the obtied result from (4.2.18) we obtain the equality

(4.2.21)  w_{1}(Δ, z) = u(z, τ_{1}) = 3 log(τ_{1}) + 3(Δ − 1) log(τ_{1} − 1)−
3(Δ + 1) log(τ_{1} + 1)−
3(Δ − 1) log(1 − δ_{0})) − log(z),

where z ≥ 1. Let

(4.2.22)  w(z, η) = u(z, (1 + η)/(1 − η)),

where 0 ≤ η < 1. Then, in view of (4.2.21) and (4.2.11)

(4.2.23)  w_{1}(Δ, z) = u(z, τ_{1}) = w(z, η_{1}^{∧}(r, π, δ_{0})) =

3 log (\frac{1 + η_{1}^{∧}(r, π, δ_{0})}{1 − η_{1}^{∧}(r, π, δ_{0})}) + 3(Δ − 1) log (\frac{1 + η_{1}^{∧}(r, π, δ_{0})}{1 − η_{1}^{∧}(r, π, δ_{0})} − 1) −

3(Δ + 1) log (\frac{1 + η_{1}^{∧}(r, π, δ_{0})}{1 − η_{1}^{∧}(r, π, δ_{0})} + 1) −

3(Δ − 1) log(1 − δ_{0})) − log(z) = 3 log(1 − (η_{1}^{∧}(r, π, δ_{0}))^{2}) − 6 log(2) + 3(Δ − 1) log(η_{1}^{∧}(r, π, δ_{0})−

18 LEONID A. GUTNIK

3(Δ − 1) log(1 − δ_{0})) − log(z).

Therefore, in view of (2.4.55),

(4.2.24)  exp(w_{1}(Δ, z)) =
−z^{−1}(η_{1}^{∧}(r, π, δ_{0})^{3(Δ−1)}((η_{1}^{∧}(r, π, δ_{0}))^{2} − 1)^{3}(1 − δ_{0})^{−3(Δ−1)}2^{−6} =
−z^{−1}h^{∼}(η_{1}^{∧}(r, π, δ_{0})) = −z^{−1}P_{1}^{∼}(η_{1}^{∧}(r, π, δ_{0}))/Q_{1}^{∼}(η_{1}^{∧}(r, π, δ_{0})).

According to (4.2.19) and (4.2.24), (in correspondence with (1.2.8) and (1.2.15))

(4.2.25) z^{ν}(R_{1}(τ_{1}+θ_{ν}/)(νΔ); ν))^{3}z^{−τ_{1}+θ_{ν}/(νΔ)} = (−h^{∼}(η_{1}^{∧}(r, π, δ_{0})))^{ν} exp(O(1)),

where z = r^{3} ≥ 1 and ν → ∞. If z ≥ 1 and ν ∈ ℕ + 2, then, in view of (4.2.12),

(4.2.26)  |(3/2) log((τ^{2} − τδ_{0})(1 − δ_{0})/(τ^{2} − 1)) − 3θ_{1}^{∗}(Δ, τ, ν)/4| ≤

(3/2) log (1 + \frac{1 + δ_{0}}{τ − δ_{0}} ) + (3/2) log (1 + \frac{1}{τ − 1}) − (3/2) log(1 − δ_{0}) + 3/4 ≤ (3/2) log (1 + \frac{1 + δ_{0}}{1 − δ_{0}} ) + (3/2) log(1 + Δν) − (3/2) log(1 − δ_{0}) + 3/4 ≤

c_{1}(Δ, z)(log(ν)),

where c_{1}(Δ, z) = ((3/2) log (1 + \frac{1 + δ_{0}}{1 − δ_{0}} ) + (3/2) log(1 + Δ) + 1 − (3/2) log(1 − δ_{0}) + 3/4. In view of (4.2.17), (4.2.26),

(4.2.27)  log(z^{−ντΔ}(R_{1}(ντΔ))^{3}) =
νw(z, η) + c_{1}(Δ, z)θ_{2}^{∗}(Δ, τ, ν)(log(ν)),

where |θ_{2}^{∗}(Δ, τ, ν)| ≤ 1. In view of (4.2.9), (4.2.10) and (4.2.22),

(4.2.28)  h(z, η_{1}^{∧}(r, π, δ_{0})) = 0,
(4.2.29)  h(z, η)(η − η_{1}^{∧}(r, π, δ_{0})) < 0,

where z ≥ 1, 0 ≤ η < 1 and η ≠ η_{1}^{∧}(r, π, δ_{0}),

(4.2.30)  − (\frac{∂}{∂η} h) (z, η_{1}^{∧}(r, π, δ_{0}) =

(1 + δ_{0})^{3}2^{−6}z^{−1}24(1 + γ_{1})^{2}(η_{1}^{∧}(r, π, δ_{0}))^{2}z^{2/3} (\frac{2z^{1/3} − 1}{1 + δ} − η_{1}^{∧}(r, π, δ_{0}) >

\frac{3}{2}z^{−1/3}(η_{1}^{∧}(r, π, δ_{0}))^{2}(2z^{1/3} − 2 + δ_{0}) > 0,

(4.2.31)  (\frac{∂}{∂τ} u) (z, τ) = 3Δ log(τ) + 3Δ − 3Δ log(τ − 1) − 3Δ−

3Δ log(τ + 1) − 3Δ + 3Δ log(τ − δ_{0}) + 3Δ−

Δ log(z) = Δ log(1 + g(z, τ)),

where τ ≥ 1, z ≥ 1,

(4.2.32)  (\frac{∂}{∂η} w) (z, η) =

2(1 − η)^{−2} (\frac{∂}{∂τ} u) (z, (1 + η)/(1 − η)) = 2(1 − η)^{−2} log(1 + h(z, η)),

where 0 ≤ η < 1, z ≥ 1,

(4.2.33)  ((\frac{∂}{∂η} )^{2} w) (z, η) =

2Δ((1 − η)^{−2}(1 + h(z, η))^{−1} (\frac{∂}{∂η} h) (z, η) + 4(1 − η)^{−3} log(1 + h(z, η)),

where 0 ≤ η < 1, z ≥ 1,

(4.2.34)  (\frac{∂}{∂η} w) (z, η_{1}^{∧}(r, π, δ_{0})) = 0,

(4.2.35)  ε_{1}(Δ, z) = −2^{−1} ((\frac{∂}{∂η} )^{2} w) (z, η_{1}^{∧}(r, π, δ_{0})) =

20 LEONID A. GUTNIK

−2Δ((1 − η_{1}^{∧}(r, π, δ_{0}))^{−2}(1 + h(z, η_{1}^{∧}(r, π, δ_{0})))^{−1} (\frac{∂}{∂η} h) (z, η_{1}^{∧}(r, π, δ_{0})) > 0,

where z ≥ 1. There exists δ_{1}(Δ, z) ∈ (0, η_{1}^{∧}(r, π, δ_{0})/2), such that

(4.2.36)  |((\frac{∂}{∂η})^{2}w)(z, η) − (\frac{∂}{∂η})^{2}w)(z, η_{1}^{∧}(r, π, δ_{0})| ≤ ε_{1}(Δ, z),

where z ≥ 1, 0 ≤ η < 1, |η − η_{1}^{∧}(r, π, δ_{0})| ≤ δ_{1}(Δ, z). Let

η_{1+k}^{∗} = η_{1}^{∧}(z^{1/3}, π, δ_{0}) + (−1)^{k}δ_{1}(Δ, z),

where k = 1, 2. Clearly, η_{1+k}^{∗} ∈ (0, 1) for k = 1, 2. Let further

τ_{1+k}^{∗} = τ_{1+k}^{∗}(Δ, z) = (1 + η_{1+k}^{∗}(r, π, δ_{0}))/(1 − η_{1+k}^{∗}(r, π, δ_{0})),

where k = 1, 2. Clearly, 1 < τ_{2}^{∗} < τ_{1}^{∗} < τ_{3}^{∗},

|τ_{1+k}^{∗} − τ_{1}^{∗}| = \frac{2δ_{1}(Δ, z)}{(1 − η_{1}^{∧}(r, π, δ_{0}))(1 − η_{1+k}^{∗}(r, π, δ_{0}))} > 2δ_{1}(Δ, z),

where k = 1, 2. If z ≥ 1, τ_{2}^{∗} ≤ τ, then instead (4.2.26) we have more strong inequality

(4.2.37)  |(3/2) log((τ^{2} − τδ_{0})(1 − δ_{0})/(τ^{2} − 1)) − 3θ_{1}^{∗}(Δ, τ, ν)/4| ≤

(3/2) log (1 + \frac{1 + δ_{0}}{τ − δ_{0}} )+(3/2) log (1 + \frac{1}{τ − 1})−(3/2) log(1−δ_{0})+3/4 ≤ c_{2}(Δ, z), where c_{2}(Δ, z) = (3/2) log (1 + \frac{1 + δ_{0}}{τ_{2} − δ_{0}} )+(3/2) log (1 + \frac{1}{τ_{2} − 1})−(3/2) log(1−δ_{0})+3/4. Therefore, if z ≥ 1, τ_{2}^{∗} ≤ τ, then instead (4.2.27) we have equality

log(z^{−ντΔ}(R_{1}(ντΔ))^{3}) =

(4.2.38)  νw_{1}(z, η) + c_{2}(Δ, z)θ_{3}^{∗}(Δ, τ, ν)(log(ν)),

where |θ_{3}^{∗}(Δ, τ, ν)| ≤ 1. In view of (4.2.32), (4.2.34), (4.2.35),(4.2.36), (4.2.38)

((\frac{∂}{∂η})^{2}w)(z, η)) ≤ −ε_{1}(Δ, z),

ON LINEAR FORMS WITH COEFFICIENTS IN ℕζ(1 + ℕ) 21

where z ≥ 1, 0 ≤ η < 1, |η − η_{1}^{∧}(r, π, δ_{0})| ≤ δ_{1}(Δ, z),

(4.2.39)  log(z^{−ντΔ}(R_{1}(ντΔ))^{3}) =
νw_{1}(z, η) + c_{2}(Δ, z)θ_{3}^{∗}(Δ, τ, ν)(log(ν)),

νw(z, η_{1}^{∧}(r, π, δ_{0})) + ν(\frac{∂}{∂η} w)(z, η_{1}^{∧}(r, π, δ_{0})(η − η_{1}^{∧}(r, π, δ_{0})+

ν((∂∂η)^{2}w)(z, η^{∼}))(η − η_{1}^{∧}(r, π, δ_{0})^{2}/2 + c_{2}(Δ, z)θ_{3}^{∗}(Δ, τ, ν) ≤ νw_{1}(Δ, z) − νε_{1}(Δ, z)(η − η_{1}^{∧}(r, π, δ_{0}))^{2}/2 + c_{2}(Δ, z)θ_{3}^{∗}(Δ, τ, ν)

where

νΔτ ∈ Δ + ℕ, 0 ≤ η < 1, |η − η_{1}^{∧}(r, π, δ_{0}| ≤ δ_{1}(Δ, z),
|η^{∼} − η_{1}^{∧}(r, π, δ_{0}| ≤ δ_{1}(Δ, z)| < |η − η_{1}^{∧}(r, π, δ_{0}| ≤ δ_{1}(Δ, z).

Therefore, since

(4.2.40)  η = (τ − 1)/(τ + 1),

it follows that

(4.2.41)  log(z^{−ντΔ}(R_{1}(ντΔ; ν))^{3}) ≤

νw_{1}(Δ, z) − 2νε_{1}(Δ, z)\frac{(τ − τ_{1})^{2}}{(τ + 1)^{2}(τ_{1} + 1)^{2}} c_{2}(Δ, z)θ_{3}^{∗}(Δ, τ, ν),

where τ_{2}^{∗} ≤ τ ≤ τ_{3}^{∗}. Let

(4.2.42)  M = sup \{|(\frac{∂}{∂τ} g) (z, τ)|: τ_{2}^{∗} ≤ τ ≤ τ_{3}^{∗}\} , α^{∗} ∈ (0, 1/2),
t_{1+k} = t_{1+k}(Δ, z, ν) ∈ ℕ, ν_{1}(Δ, z) ∈ ℕ + 2,

where k = 1, 2, z ≥ 1, and let, if ν ∈ ν_{1}(Δ, z) + ℕ − 1, z ≥ 1, then

(4.2.43)  ν^{−α^{∗}} < δ_{1}(Δ, z)/2, Mν^{−α^{∗}} < 1, ν(τ_{2}^{∗} − 1) > 1,
(4.2.44)  νΔ < ντ_{2}^{∗} < ντ_{1} − νΔ2δ_{1}(Δ_{1}, z) < ντ_{1} − 2νΔν^{−α^{∗}} < t_{2}(z, ν) <

ντ_{1} −νΔν^{−α^{∗}} < ντ_{1} +νΔν^{−α^{∗}} < t_{3}(z, ν) < ντ_{1} +2νΔν^{−α^{∗}} < ντ_{1}2δ_{1}(Δ_{1}, z) < ντ_{3}^{∗}.

22 LEONID A. GUTNIK

In view of (4.2.41) and (4.2.44), if z ≥ 1, ν ∈ ν_{1}(Δ, z) + ℕ − 1,

t ∈ ([νΔ + 1, τ_{2}] ∪ [τ_{3}, ∞) ∩ ℤ, c_{3}(Δ, z) = exp(2c_{2}(Δ, z)),

then

(4.2.45)  z^{−t}(R_{1}(t; ν))^{3}) ≤ (z^{−t_{2}}(R_{1}(t_{2}; ν))^{3} + z^{−t_{3}}(R_{1}(t_{3}; ν))^{3})) ≤
exp(w_{1}(Δ, z)ν) exp(−2ε_{1}(Δ, z)(τ_{3} + 1)^{−4}ν^{1−2α^{∗}})c_{3}(Δ, z).

Therefore in view of (4.2.24) (and, as in (4.2.4),),

(4.2.46)  z^{ν} (( ∑_{t=νΔ+1}^{t_{2}(z,ν)}(R_{0}(t, ν))^{3}z^{−t}) + ( ∑_{t_{3}(z,ν)}^{∞}(R_{0}(t, ν))^{3}z^{−t})) ≤
2^{−3}(νΔ)^{−2}(−h^{∼}(η_{1}^{∧}(r, π, δ_{0})))^{ν}
exp(−2ε_{1}(Δ, z)(τ_{3} + 1)^{−4}ν^{1−2α^{∗}})c_{3}(Δ, z).

In view of (4.2.14) and (4.2.25),

(4.2.47)  (ν)^{−3}(Δ(τ_{1} + 1) + 1)^{−3}(−h^{∼}(η_{1}^{∧}(r, π, δ_{0})))^{ν} exp(O(1)) ≤
z^{ν}z^{−νΔτ_{1}−θ_{ν}} (νΔτ_{1} + θ_{ν} + νΔ)^{−3}(R_{1}(νΔτ_{1} + θ_{ν}; ν))^{3} =
z^{ν}z^{[−νΔτ_{1}]}(−[−νΔτ_{1}] + νΔ)^{−3}R_{1}(−[−νΔτ_{1}]; ν)^{3} ≤

z^{ν} ∑_{t=1+νΔ}^{+∞} z^{−t}R_{0}(t; ν)^{3} = |f_{2}^{∗}(z, ν)| ≤

z^{ν}2^{−3}(νΔ)^{−2}z^{−νΔτ_{1}−θ_{ν}} (R_{1}(νΔτ_{1} + θ_{ν}; ν))^{3} ≤ 2^{−3}(νΔ)^{−2}(−h^{∼}(η_{1}^{∧}(r, π, δ_{0})))^{ν} exp(O(1)).

Let γ^{∗} = γ^{∗}(Δ, z) = 2ε_{1}(Δ, z)(τ_{3} + 1)^{−4}. In view of (4.2.46) and (4.2.47),

(4.2.48)  (f_{2}^{∗}(z, ν))^{−1}×

(−z)^{ν} (( ∑_{t=νΔ+1}^{t_{2}(z,ν)}(R_{0}(t, ν))^{3}z^{−t}) + ( ∑_{t_{3}(z,ν)}^{∞}(R_{0}(t, ν))^{3}z^{−t})) =

O(1)ν exp(−γ^{∗}ν^{1−2α^{∗}}.

ON LINEAR FORMS WITH COEFFICIENTS IN ℕζ(1 + ℕ) 23

Therefore

(4.2.49)  (−z)^{ν} ( ∑_{t_{2}(z,ν)+1}^{t_{3}(z,ν)−1}(R_{0}(t, ν))^{3}z^{−t}) =
f_{2}^{∗}(z, ν) (1 + O(1)ν exp(−γ^{∗}ν^{1−2α^{∗}}) .

Since

(4.2.50)  ((\frac{∂}{∂t}) (R_{0})^{3}) (t, ν) = (R_{0}(t, ν))^{3}×

(( ∑_{κ=ν+1}^{νΔ}(t − κ)^{−1}) − (∑_{κ=0}^{νΔ}(t + κ)^{−1})) ,

(4.2.51)  ((\frac{∂}{∂t})^{2} (R_{0})^{3}) (t, ν) = (R_{0}(t, ν))^{3}×

(( ∑_{κ=ν+1}^{νΔ}(t − 1/κ)^{−1}) − (∑_{κ=0}^{νΔ}(t + κ)^{−1}))^{2} −

(R_{0}(t, ν))^{3}×

(( ∑_{κ=ν+1}^{νΔ}(t − κ)^{−2}) − (∑_{κ=0}^{νΔ}(t + κ)^{−2})) ,

where ν ∈ ℕ − 1, it follows that

(4.2.52)  |((\frac{∂}{∂t}) (R_{0})^{3}) (t, ν)| ≤

3(R_{0}(t, ν))^{3} (((t − νΔ)^{−1} + log (1 + \frac{ν(Δ − 1) − 1}{t − νΔ})) |_{t=νΔ+1}) +

(R_{0}(t, ν))^{3} ((t^{−1} + log (1 + \frac{νΔ}{t})) |_{t=νΔ+1}) ≤ (R_{0}(t, ν))^{3}3 log(6νΔ),

where t ≥ νΔ + 1 and ν ∈ ℕ,

(4.2.53)  |((\frac{∂}{∂t}) (R_{0})^{3}) (t, ν)| ≤ (R_{0}(t, ν))^{3}(9(log(6νΔ))^{2} + 5),

where t ≥ νΔ + 1 and ν ∈ ℕ. If t ≥ νΔτ_{2}^{∗}, ν ∈ ℕ, then, using (4.2.50), (4.2.7), (and applying, for example, Sonin’s formula, [20], questions in the end of the chapter II, the question 8) we obtain the equality

(4.2.54)  ((\frac{∂}{∂t}) (R_{0})^{3}) (t, ν) = 3(R_{0}(t, ν))^{3}×

(log (\frac{(t − ν)t}{t^{2} − ν^{2}Δ^{2}} ) + \frac{O(1)}{νΔ(τ_{2}^{∗} − 1)}) =

3(R_{0}(t, ν))^{3} (log (\frac{(τ − δ_{0})τ}{τ^{2} − 1} ) + \frac{O(1)}{νΔ(τ_{2}^{∗} − 1)}) =

(R_{0}(t, ν))^{3} (log(z(1 + g(z, τ)) + \frac{O(1)}{νΔ(τ_{2}^{∗} − 1)})

and

(4.2.55)  ((\frac{∂}{∂t})^{2} (R_{0})^{3}) (t, ν) =

(R_{0}(t, ν))^{3} ((log(z(1 + g(z, τ)) + \frac{O(1)}{νΔ(τ_{2}^{∗} − 1)})^{2} + \frac{O(1)}{νΔ(τ_{2}^{∗} − 1)}) .

If t = νΔτ ∈ [t_{2}, t_{3}], then, according to (4.2.44), |τ − τ_{1}| < 2ν^{−α^{∗}} and, since g(z, τ_{1}) = 0, in view of (4.2.42)

(4.2.56)  log(z(1 + g(z, τ) = O(nu^{−α^{∗}}).

Therefore,according to (4.2.54) and (4.2.55)

(4.2.57)  ((\frac{∂}{∂t}) (R_{0})^{3}) (t, ν) = (R_{0}(t, ν))^{3}(log(z) + O(nu^{−α^{∗}}),

(4.2.58) ((\frac{∂}{∂t})^{2} (R_{0})^{3}) (t, ν) = (R_{0}(t, ν))^{3}((log(z))^{2} + (1 + log(z))O(nu^{−α^{∗}}).

In view of (4.2.48), (4.2.52) and (4.2.53),

(4.2.59)  (f_{2}^{∗}(z, ν))^{−1}(−z)^{ν}(( ∑_{t=νΔ+1}^{t_{2}(z,ν)} (\frac{∂}{∂t}(R_{0})^{3}) (t, ν)z^{−t}) +

ON LINEAR FORMS WITH COEFFICIENTS IN ℕζ(1 + ℕ) 25

(f_{2}^{∗}(z, ν))^{−1}(−z)^{ν} ( ∑_{t_{3}(z,ν)}^{∞} (\frac{∂}{∂t}(R_{0})^{3}) (t, ν)z^{−t}) =

O(1)ν exp(−γ^{a}stν^{1−2α^{a}st}(log(6ν)),

(4.2.60)  (f_{2}^{∗}(z, ν))^{−1}(−z^{ν})(( ∑_{t=νΔ+1}^{t_{2}(z,ν)} ((\frac{∂}{∂t})^{2} (R_{0})^{3}) (t, ν)z^{−t}) +

(f_{2}^{∗}(z, ν))^{−1}(−z)^{ν} ( ∑_{t_{3}(z,ν)}^{∞} ((\frac{∂}{∂t})^{2} (R_{0})^{3}) (t, ν)z^{−t}) =

O(1)ν exp(−γ^{a}stν^{1−2α^{a}st}(log(6ν))^{2},

where ν ∈ ν_{1}(Δ, z) + ℕ − 1. In view of (4.2.57) and (4.2.58),

(4.2.61)  (−z)^{ν} ( ∑_{t_{2}(z,ν)+1}^{t_{3}(z,ν)−1} (\frac{∂}{∂t}(R_{0})^{3}) (t, ν)z^{−t}) =

(−z)^{ν} ( ∑_{t_{2}(z,ν)+1}^{t_{3}(z,ν)−1}(R_{0}(t, ν))^{3}z^{−t}) (log(z) + O(nu^{−α^{∗}}),

(4.2.62)  (−z)^{ν} ( ∑_{t_{2}(z,ν)+1}^{t_{3}(z,ν)−1} ((\frac{∂}{∂t})^{2} (R_{0})^{3}) (t, ν)z^{−t}) =

(−z)^{ν} ( ∑_{t_{2}(z,ν)+1}^{t_{3}(z,ν)−1}(R_{0}(t, ν))^{3}z^{−t}) ((log(z))^{2} + (1 + log(z))O(nu^{−α^{∗}}),

where ν ∈ ν_{1}(Δ, z) + ℕ − 1, z ≥ 1. In view of (4.2.49), (4.2.61) and (4.2.62),

(4.2.63)  (−z)^{ν} ( ∑_{t_{2}(z,ν)+1}^{t_{3}(z,ν)−1} (\frac{∂}{∂t}(R_{0})^{3}) (t, ν)z^{−t}) =
f_{2}^{∗}(z, ν) (1 + O(1)ν exp(−γ^{a}stν^{1−2α^{∗}}) (log(z) + O(ν^{−α^{∗}}),
(4.2.64)  (−z)^{ν} ( ∑_{t_{2}(z,ν)+1}^{t_{3}(z,ν)−1} ((\frac{∂}{∂t})^{2} (R_{0})^{3}) (t, ν)z^{−t}) =

f_{2}^{∗}(z, ν) (1 + O(1)ν exp(−γ^{a}stν^{1−2α^{a}st}) ((log(z))^{2} + (1 + log(z))O(ν^{−α^{∗}}), where ν ∈ ν_{1}(Δ, z)+ℕ−1, z ≥ 1. The assertion of the Lemma follows from (4.2.47), (4.2.59), (4.2.63), (4.2.60), (4.2.64). ■

§3 A correction to [38].

In (2.3.16) of paper [38], there is a misprint.

The correct equality is:

α_{j} = (log(r_{j}^{∧}R_{1}/R_{2}))/log(R_{2})

(instead of the incorrect one printed there: α_{j} = (log(r_{j}^{∧}R_{1}))/log(R_{2})).

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Moscow State Institute of Electronics and Mathematics Bolshoi Trekhsvyatitelskij pereulok, 1-3/12, str.8 Moscow 109028 Russian Federation E-mail: gutnik@gutnik.mccme.ru