ON LINEAR FORMS WITH COEFFICIENTS IN ℕζ(1 + ℕ) The detailed version, Part 7

L.A.Gutnik

In memory of Professor Alexandr Buchstab

Table of contents

§0. Foreword. §1. Proof of the Theorem. §2. Verification of the previous calculations on computer and

the values of γ^{∗}(Δ) for other Δ. §3. Epilogue.

§0. Foreword.

Here I complete the proof of the following theorem.

Theorem Let γ = 43.464412,

ϕ_{i}(x_{1}, x_{2}) = ∑_{k=1}^{2}(2ik − i − k + 2) ζ(i + k + 1) x_{k}, i = 1, 2.

For any d ∈ ℝ, let ∥d∥ denote the distance between d and ℤ. There exists a positive constant c such that

∥ϕ_{1}(x_{1}, x_{2})∥ + ∥ϕ_{2}(x_{1}, x_{2})∥ ≥ c(|x_{1}| + |x_{2}|)^{−γ},

as soon as x_{1} ∈ ℤ, x_{2} ∈ ℤ, and |x_{1}| + |x_{2}| > 0.

Corollary. If p ∈ ℚ, q ∈ ℚ, p^{2} + q^{2} > 0, then

{2pζ(3) + 3qζ(4), 3pζ(4) + 6qζ(5)} ⊄ ℚ

and therefore

{ζ(3 + 2k)/ζ(4), (12ζ(3)ζ(5) − 9(ζ(4))^{2})/ζ(4)} ⊄ ℚ,

where k = 0, 1.

L.A.Gutnik, On linear forms with coefficients in ℕζ(⊮ + ℕ), Part 7. 2

Acknowledgments. I express my deepest thanks to Professors B.Z.Moroz, I.I. Piatetski-Shapiro, A.G.Aleksandrov, P.Bundshuh and S.G.Gindikin for help and support.

§1. Proof of the Theorem.

Let

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

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

(3) 2γ_{ν,k}^{∗} = 9α_{ν,k}^{∗} (− ( ∑_{κ=ν+k+1}^{νΔ+k} 1/κ^{2}) − ( ∑_{κ=1}^{νΔ−k} 1/κ^{2}) + (∑_{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.

α^{∗}(z; ν) = (−z)^{ν} ∑_{k=0}^{νΔ} α_{ν,k}^{∗}z^{k} = f_{1}^{∗}(z; ν), (4)

β^{∗}(z; ν) = (−z)^{ν} ∑_{k=0}^{νΔ} β_{ν,k}^{∗}z^{k} (5)

γ^{∗}(z; ν) = (−z)^{ν} ∑_{k=0}^{νΔ} γ_{ν,k}^{∗}z^{k}, (6)

ϕ^{∗}(z; ν) = (−z)^{ν} ∑_{k=0}^{νΔ} α_{ν,k}^{∗}z^{k} ∑_{t=1}^{k+ν} z^{−t}t^{−3} + (7)

+ (−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};

ψ^{∗}(z; ν) = (−z)^{ν} ∑_{k=0}^{νΔ} 3α_{ν,k}^{∗} z^{k} ∑_{t=1}^{k+ν} z^{−t}t^{−4} + (8)

+ (−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} =

= (−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};

ξ^{∗}(z; ν) = (−z)^{ν} ∑_{k=0}^{νΔ} 6α_{ν,k}^{∗}z^{k} ∑_{t=1}^{k+ν} z^{−t}t^{−5} + (9)

+ (−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}.

For a prime p ∈ ℕ, let v_{p} stand for the p-adic valuation on ℚ, and π(x) denotes the number of prime p ∈ ℕ∩(−∞, x], where x ∈ ℝ. So, π(x) = 0 for x < 2. Let {α, β, γ} ⊂ ℝ. If π(α) < π(β) and 1 ≤ γ, then let d_{α,β,γ} denotes the smallest positive integer d satisfying the condition

v_{p} (\frac{d}{κ}) ≥ 0 (10)

L.A.Gutnik, On linear forms with coefficients in ℕζ(⊮ + ℕ), Part 7. 4

for all the primes p ∈ ℕ ∩ (α, β] and all the κ ∈ ℕ ∩ [1, γ]. We let, by definition, d_{α,β,γ} = 1 if either γ < 1, or π(β) ≤ π(α). Clearly,

d_{α,β,γ} = ∏_{α<p≤β} p^{[ln(γ)/ln(p)]} ≤ γ^{max(π(β)−π(α),0)}. (11)

Lemma 7.1. If 1 ≤ α ≤ β, 1 ≤ γ ≤ γ_{0}α for some γ_{0} > 0, then for any ε > 0 there exists C_{0}(γ_{0}, ε) satisfying the following inequalities: d_{α,β,γ} ≤ C_{0}(γ_{0}, ε) exp(α(β/α − 1 + ε)), (12) and d_{1,α,γ} ≤ C_{0}(γ_{1}, ε) exp(α(1 + ε)). (13) Proof (see [64], Lemma 4). It may be assumed that β ≤ γ; then it follows from the inequalities (11) – (13) that d_{α,β,γ} ≤ (14)

exp((ln(γ_{0}) + ln(α))×
((β − α)/ ln(eα) + O(β − α)/ ln^{2}(eα)) ≤
C_{0}(γ_{0}, ε) exp(α(β/α − 1 + ε)),

d_{1,α,γ} ≤ (15)

exp((ln(γ_{1}) + ln(α))(α/ ln(eα) + O(α/ ln^{2}(eα))) ≤
C_{0}(γ_{0}, ε) exp((α(1 + ε)). ■

Lemma 7.2. If ν > 2/Δ, (16)

p > νΔ, (17) then v_{p}(β_{ν,k}^{∗}) ≥ 0 (18)

v_{p}(γ_{ν,k}^{∗}) ≥ 0 (19)

Proof (see [58], p.23). If p > νΔ + k, then (18) – (19) directly follow from the equalities (1) – (3). If νΔ < p ≤ νΔ + k, then we must consider in the expressions (2) – (3) only the case ν +k < κ ≤ νΔ+k; if νΔ < p ≤ ν +k, then we have

p < κ ≤ νΔ + k ≤ 2νΔ < 2p

and, consequently, (κ, p) = 1; therefore in this case we must consider only p > max(ν + k, νΔ).

But then v_{p}(κ) ≤ 1,

v_{p} (\binom{νΔ + k}{νΔ − ν}^{3}) ≥ 3,

v_{p} (κ^{−2}\binom{νΔ + k}{νΔ − ν}^{3}) ≥ 1.

■ According to the Lemma 7.2,

d_{1,νΔ,2νΔ}β_{ν,k}^{∗} ∈ ℤ, (20)

(d_{1,νΔ,2νΔ})^{2}γ_{ν,k}^{∗} ∈ ℤ. (21)

Let ν ∈ ℕ, Δ ∈ ℕ, Δ ≥ 2. Let us consider the numbers

\binom{νΔ}{k}\binom{νΔ + k}{νΔ − ν}, (22)

where k = 0, …, νΔ. How small may be chosen r(ν) ∈ (0, +∞) ∩ ℚ such that

r(ν)\binom{νΔ}{k}\binom{νΔ + k}{νΔ − ν} ∈ ℤ

for all the k = 0, …, νΔ? Probably G.V. Chudnovsky was the first man, who discovered, that r(ν) may be chosen sufficiently small; Hata ([17]) in details studied this effect. Therefore I name such r(ν) by Chudnovsky-Hata’s multiplier.

Lemma 7.3. Let p ∈ ℕ is a prime number such that

2{νΔ/p} < {ν(Δ − 1)/p}. (23)

Then

v_{p} (\binom{νΔ}{k}\binom{νΔ + k}{νΔ − ν}) ≥ 1, (24)

where k = 0, …, νΔ.

Proof. See [77], section 2. ■

Let d_{1}^{∗}(ν) denotes the product of all the prime numbers p ∈ ℕ, which satisfy to the condition (23). Then, according to the Lemma 7.3,

(d_{1}^{∗}(ν))^{−3}α_{ν,k}^{∗} ∈ ℤ (25)

for any k = 0, …, νΔ.

Lemma 7.4. If (16) holds, then

d_{1,νΔ,2νΔ}(d_{1}^{∗}(ν))^{−3}β_{ν,k}^{∗} ∈ ℤ, (26)

(d_{1,νΔ,2νΔ})^{2}(d_{1}^{∗}(ν))^{−3}γ_{ν,k}^{∗} ∈ ℤ, (27)

where k = 0, …, νΔ.

Proof. We prove that

v_{p} (d_{1,νΔ,2νΔ}(d_{1}^{∗}(ν))^{−3}β_{ν,k}^{∗} ∈ ℤ) ≥ 0, (28)

v_{p} ((d_{1,νΔ,2νΔ})^{2}(d_{1}^{∗}(ν))^{−3}γ_{ν,k}^{∗}) ≥ 0 (29)

for any prime number p ∈ ℕ.

We partially repeat the proof of the Lemma 7.2. If p > νΔ + k, then, since κ ≤ νΔ + k, it follows that v_{p}(κ) = 0, and, according to (25), the relations (28) – (29) hold. If p > νΔ, then, since κ ≤ νΔ + k ≤ 2νΔ < 2p, it follows that v_{p}(κ) ≤ 1; moreover, if p ≤ ν + k < κ, then, since κ < 2p, it follows that v_{p}(κ) = 0, and, in view of (25), the relations (28) – (29) hold. On the other hand, if p > max(νΔ, ν + k) then the inequality (23) turns into equality with both sides equal to zero. Therefore v_{p}(d_{1}^{∗}(ν)) = 0 and the relations (18) – (19) hold. Finally if p ≤ νΔ, then, since κ ≤ 2νδ, it follows that v_{p}(d_{1,νΔ,2νΔ}/κ) ≥ 0 and, according to (25), the relations (18) – (19) hold. ■

Let

D^{∗∗}(ν) = d_{1,ν(Δ+1),2νΔ}. (30)

Then d_{1,νΔ,2νΔ} is divisor of D^{∗∗}(ν) and according to the Lemma 7.4,

D^{∗∗}(ν)(d_{1}^{∗}(ν))^{−3}β_{ν,k}^{∗} ∈ ℤ, (31)

(D^{∗∗}(ν))^{2}(d_{1}^{∗}(ν))^{−3}γ_{ν,k}^{∗} ∈ ℤ, (32)

where k = 0, …, νΔ.

Lemma 7.5. If (16) holds, then

(D^{∗∗}(ν))^{5}(d_{1}^{∗}(ν))^{−3}α^{∗}(z; ν) ∈ ℤ[z], (33)

(D^{∗∗}(ν))^{5}(d_{1}^{∗}(ν))^{−3}β^{∗}(z; ν) ∈ ℤ[z], (34)

(D^{∗∗}(ν))^{5}(d_{1}^{∗}(ν))^{−3}γ^{∗}(z; ν) ∈ ℤ[z], (35)

(D^{∗∗}(ν))^{5}(d_{1}^{∗}(ν))^{−3}ϕ^{∗}(z; ν) ∈ ℤ[z], (36)

(D^{∗∗}(ν))^{5}(d_{1}^{∗}(ν))^{−3}ψ^{∗}(z; ν) ∈ ℤ[z], (37)

(D^{∗∗}(ν))^{5}(d_{1}^{∗}(ν))^{−3}ξ^{∗}(z; ν) ∈ ℤ[z]. (38)

Proof. The relations (33) – (34) directly follow from (4) – (6), (31) – (32) and (25). It follows from (7) – (9), (25), (31) – (32) and (30) that

(−1)^{ν}(D^{∗∗}(ν))^{3}(d_{1}^{∗}(ν))^{−3}ϕ^{∗}(z; ν) = (39)

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

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

∑_{k=0}^{νΔ}(D^{∗∗}(ν))^{2}(d_{1}^{∗}(ν))^{−3}γ_{ν,k}^{∗}z^{k+ν−t} ∑_{t=1}^{k+ν}(D^{∗∗}(ν))t^{−1} ∈ ℤ[z],

(−1)^{ν}(D^{∗∗}(ν))^{4}(d_{1}^{∗}(ν))^{−3}ψ^{∗}(z; ν) = (40)

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

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

∑_{k=0}^{νΔ}(D^{∗∗}(ν))^{2}(d_{1}^{∗}(ν))^{−3}γ_{ν,k}^{∗} ∑_{t=1}^{k+ν} z^{k+ν−t}(D^{∗∗}(ν))^{2}t^{−2} ∈ ℤ[z],

(−1)^{ν}(D^{∗∗}(ν))^{5}(d_{1}^{∗}(ν))^{−3}ξ^{∗}(z; ν) = (41)

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

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

(−z)^{ν} ∑_{k=0}^{νΔ}(D^{∗∗}(ν))^{2}(d_{1}^{∗}(ν))^{−3}γ_{ν,k}^{∗} ∑_{t=1}^{k+ν} z^{k+ν−t}(D^{∗∗}(ν))^{3}t^{−3} ∈ ℤ[z].

■

Though in the initial work I used computer for calculation, I shall make here all calculations ’by hands’ using only calcuator of the firm ’Casio.’ Therefore everyone can check them up. I take below Δ = 13. According to (13) and (30),

(Δ + 1 − ε)ν − O(1) ≤ ln(D^{∗∗}(ν)) ≤ (Δ + 1 + ε)ν + O(1), (42)

with any ε > 0 and O(1) depending only from ε. For Δ = 13 we have

(14 − ε)ν − O(1) ≤ ln(D^{∗∗}(ν)) ≤ (14 + ε)ν + O(1).

In view of (32) and (45) in [77],

ν(ΔI − ε) + O(1) ≤ log(d_{1}^{∗}(ν)) ≤ ν(ΔI + ε) + O(1),

where

I = I(Δ) = ln(Δ) − \frac{Δ − 1}{2Δ} ln(Δ − 1) − \frac{Δ + 1}{2Δ} ln(Δ + 1)−

L.A.Gutnik, On linear forms with coefficients in ℕζ(⊮ + ℕ), Part 7. 8

\frac{π}{2Δ} ∑_{1≤κ≤(Δ−1)/2} cot(πκ/(Δ − 1))+

\frac{π}{2Δ} ∑_{1≤κ≤(Δ+1)/2} cot(πκ/(Δ + 1)).

For Δ = 13 we have

I = ln(13) − \frac{6}{13} ln(12) − \frac{7}{13} ln(14)−

\frac{π}{26} ∑_{1≤κ≤6} cot(πκ/12))+

\frac{π}{26} ∑_{1≤κ≤7} cot(πκ/14).

Further we have

ln(12) = 2, 48490665...; ln(13) = 2, 564949357...; ln(14) = 2, 63905733...;

\frac{6}{13} ln(12) = 1, 146879992...; \frac{7}{13} ln(14) = 1, 42103087...;

cot(π/12) + cot(2π/12) + cot(3π/12) + cot(4π/12) + cot(5π/12)+ cot(6π/12) = 2/ sin(pi/6) + 2/ sin(pi/3) + 1 = 5 + 4/\sqrt{3} = 7.309401077...;

2/ sin(π/7) = 4, 609529742; cot(π/14) + cot(2π/14) + cot(3π/14)+

cot(4π/14) + cot(5π/14) + cot(6π/14) + cot(7π/14) =
2/ sin(π/7) + 2/ sin(2π/7) + 2/ sin(3π/7) =
2/ sin(π/7) + 4 sin(5π/14) cos(π/14)/(sin(2π/7) cos(π/14)) =

2/ sin(π/7) + 4cos(π/7)/ sin(2π/7) = 4/ sin(π/7) = 9, 219059484...;

\frac{π}{26} ∑_{1≤κ≤7} cot(πκ/4)−

\frac{π}{26} ∑_{1≤κ≤6} cot(πκ/12) =

\frac{π}{26}(9, 219059484... − 7, 309401077...) =

(9, 219059484... − 7, 309401077...) = (1, 909658407...)π/26 = 0, 230744954...

I = ln(13) − \frac{6}{13} ln(12) − \frac{7}{13} ln(14)−

\frac{π}{26} ∑_{1≤κ≤6} cot(πκ/12))+

\frac{π}{26} ∑_{1≤κ≤7} cot(πκ/14) =

2, 564949357... − 1, 146879992... − 1, 42103087... + 0, 230744954... = = 0.227783449... . 3ΔI = 39 × 0, 227783449... = ...8, 883554511... ,

L.A.Gutnik, On linear forms with coefficients in ℕζ(⊮ + ℕ), Part 7. 9

Let ω_{2}(Δ) = 5(Δ + 1) − 3ΔI(Δ) = (43)

5(Δ + 1) − \frac{3}{2}(Δ − 1) ln (\frac{Δ}{Δ − 1}) +

\frac{3}{2}(Δ + 1) ln (\frac{Δ + 1}{Δ}) +

\frac{3π}{2} ∑_{1≤κ≤(Δ−1)/2} cot(πκ/(Δ − 1))−

\frac{3π}{2} ∑_{1≤κ<(Δ+1)/2} cot(πκ/(Δ + 1)).

Then ω_{2}(13) = 70 − 8, 883554511... = 61.11644549..., (44) and ν(ω_{2}(13) − ε) − O(1) ≤ ln((D^{∗∗}(ν))^{5}/(d_{1}^{∗}(ν)^{3}) ≤ (45)

ν(ω_{2}(13) + ε) + O(1)

with any ε > 0 and O(1) depending only from ε.

In view of (5.27) in [71], we must calculate the value |h^{∼}(η_{1}^{∧}(1, π/3, δ_{0})|; we compute also the value |h^{∼}(η_{0}^{∧}(1, π/3, δ_{0})|. According to (3.1.43) and (3.1.71) in [62], for our case Δ = 13 we have q_{1} = 36/49 = 0, 734693877..., q_{2} = 676/49 = 13, 79591837... (46)

q_{0} = 12 × 169/49 = 41, 3877551... . (47) In view of (3.1.10) in [62], |D_{0}(1, π/3, 1/13)|^{2} = 2 + (1/676)^{2} − 1/676 + 1 + 1/676 = 3 + (1/676)^{2} =

1370929/(676)^{2} = 3, 000002188... ,

|D_{0}(1, π/3, 1/13)| = 1170, 866773.../676 = 1, 732051439... .

In view of (3.1.41) in [62],

p_{1} = 8(|R_{0}^{∗}(r, ψ, δ_{0})|^{2} + |R_{0}(r, ψ, δ_{0})|^{2})/(1 + δ_{0})^{2} =

8 (r^{2} + rt + \frac{1}{4} + |D_{0}(r, ψ, δ_{0})|) /(1 + δ_{0})^{2},

and in our case p_{1} = 8 (7/4 + |D_{0}(1, π/3, 1/3)|) /(1 + 1/13)^{2} = (48)

(169/196)8 (1, 75 + 1, 732051439...) = 24.0190487... .

In view of (3.1.42) in [62],

p_{2} = 8 (1 − \frac{1}{26} + \frac{1}{676} + |D_{0}(1, π/3, 1/3)|) /(1 + 1/13)^{2} = (49)

(169/196)8 (651/676 + 1, 732051439...) = 18, 59047727... .

In view of (3.1.70) and (3.1.61) in [62],

p_{0} = 8(|R_{−1}^{∗}(r, ψ, δ_{0})|^{2} + |R_{−1}(r, ψ, δ_{0})|^{2})/(1 + δ_{0})^{2} =

8 (r^{2} + (2 + δ_{0})^{2}/4 + r(2 + δ_{0}) cos(ψ) + |D_{0}(r, ψ, δ_{0})|) /(1 + δ_{0})^{2},

and in our case p_{0} = 8 (2107/676 + |D_{0}(1, π/3, 1/3)|) /(1 + 1/13)^{2} = (50)

(169/196)8 (3, 116863905... + 1, 732051439...) = 33, 44762013... .

Further we have

p_{1}/2 = 24, 0190487.../2 = 12, 00952435... ,
p_{2}/2 = 18, 59047727.../2 = 9, 295238635... ,
p_{0}/2 = 33, 44762013.../2 = 16, 72381007... ,

\sqrt{(p_{1}/2)^{2} − q_{1}} = \sqrt{(12, 00952435...)^{2} − 0, 734693877...} = 11, 97889733... , \sqrt{(p_{2}/2)^{2} − q_{2}} = \sqrt{(9, 295238635...)^{2} − 13, 79591837...} = 8, 520888622... , \sqrt{(p_{0}/2)^{2} − q_{0}} = \sqrt{(16, 72381007...)^{2} − 41, 3877551...} = 15, 43690604... .

In view of (3.1.37) in [62], |η_{1}(1, π/3, 1/13)|^{2} = p_{1}/2 − \sqrt{(p_{1}/2)^{2} − q_{1}} = (51)

12, 00952435... − 11, 97889733... = 0, 03062702...

|η_{0}(1, π/3, 1/13)|^{2} = p_{1}/2 + \sqrt{(p_{1}/2)^{2} − q_{1}} = (52)

12, 00952435... + 11, 97889733... = 23, 98842168...

In view of (3.1.37) in [62], |η_{1}(1, π/3, 1/13) + 1|^{2} = p_{2}/2 − \sqrt{(p_{2}/2)^{2} − q_{2}} = (53)

9, 295238635... − 8, 520888622... = 0, 774350013...

|η_{0}(1, π/3, 1/13) + 1|^{2} = p_{2}/2 + \sqrt{(p_{2}/2)^{2} − q_{2}} = (54)

9, 295238635... + 8, 520888622... = 17, 816127257...

In view of (3.1.61) in [62], in our case

s = cos(ψ/2) = cos(π/6) = \frac{\sqrt{3}}{2} > δ_{0}/4 = \frac{1}{52}.

L.A.Gutnik, On linear forms with coefficients in ℕζ(⊮ + ℕ), Part 7. 11

Therefore, in view of (3.1.65) in [62], |η_{1}(1, π/3, 1/13) − 1|^{2} = p_{0}/2 − \sqrt{(p_{0}/2)^{2} − q_{0}} = (55)

16, 72381007... − 15, 43690604... = 1, 28690403... ,

|η_{0}(1, π/3, 1/13) − 1|^{2} = p_{0}/2 + \sqrt{(p_{0}/2)^{2} − q_{0}} = (56)

16, 72381007... + 15, 43690604... = 32, 16071611...

Before to go further, let us check the previous calculations.

|η_{0}(1, π/3, 1/13)|^{2}|η_{1}(1, π/3, 1/13)|^{2} =

23, 98842168... × 0, 03062702.. = 0.73469387...,

what is almost equal to above q_{1} = 36/49 = 0, 734693877... ,

|η_{0}(1, π/3, 1/13) + 1|^{2}|η_{1}(1, π/3, 1/13) + 1|^{2} =

17, 816127257... × 0, 774350013... = 13.79591837...,

what is (almost) equal to above q_{2} = 13, 79591837... , |η_{0}(1, π/3, 1/13) − 1|^{2}|η_{1}(1, π/3, 1/13) − 1|^{2} =

32, 16071611... × 1, 28690403... = 41, 38775517...,

what is almost equal to above q_{0} = 41, 3877551... , |η_{1}(1, π/3, 1/13) + 1|^{2} + |η_{1}(1, π/3, 1/13) − 1|^{2} =

0, 774350013... + 1.28690403... = 2, 061254043... ,

what ”almost” is equal to

2(1 + |η_{1}(1, π/3, 1/13)|^{2}) = 2(1 + 0, 03062702...) = 2, 06125404...;

|η_{0}(1, π/3, 1/13) + 1|^{2} + |η_{0}(1, π/3, 1/13) − 1|^{2} =
17, 816127257... + 32, 16071611... = 49, 97684336... ,

what ”almost” is equal to

2(1 + |η_{0}(1, π/3, 1/13)|^{2}) = 2(1 + 23, 98842168...) = 49, 97684336...; In view of (3.1.28) in [62],

ln(|h^{∼}(η_{1}(1, π/3, 1/13)) =
1, 5 ln(|(η_{1}(1, π/3, 1/13))^{2} − 1|^{2}) + 36 ln(13/12) − 6 ln 2+
18 ln(|η_{1}(1, π/3, 1/13)|^{2}).

Further we have 1, 5 ln(|(η_{1}(1, π/3, 1/13)) − 1|^{2} = 1, 5 ln(0, 774350013...) = −0, 383596941..., 1, 5 ln(|(η_{1}(1, π/3, 1/13)) + 1|^{2} = 1, 5 ln(1.28690403...) = 0, 378359035...,

L.A.Gutnik, On linear forms with coefficients in ℕζ(⊮ + ℕ), Part 7. 12

36 ln(13/12) = 2, 881537476...
18 ln(|η_{1}(1, π/3, 1/13)|^{2}) = 18 ln(0, 03062702...) = −62.74570776... ,
ln(64) = 4.158883083...

− ln(|h^{∼}(η_{1}(1, π/3, 1/13)|) = (57)

4, 158883083... + 62, 74570776... + 0, 383596941...−
0, 378359035... − 2, 881537476 = 64.02829127... .

Let γ_{1}^{∗}(Δ) = − ln(|h^{∼}(η_{1}(1, π/3, 1/Delta)|) − ω_{2}(Δ) (58) I view of (44) and (57), γ_{1}^{∗}(13) = 64.02829127... − 61.11644549... = (59)

2, 911845783... > 0,

and − ν(γ_{1}^{∗}(13) + ε) + O(1) ≤ (60)

ln(|h^{∼}(η_{1}(1, π/3, 1/13))|^{ν}(D^{∗∗}(ν))^{5}/(d_{1}^{∗}(ν)^{3})) ≤
ν(61.11644549... − 64.02829127... + ε) + O(1) =
−ν(γ_{1}^{∗}(13) − ε) + O(1) → −∞,

if 0 < ε < γ_{1}^{∗}(13)/2 and ν → +∞.

Now we compute also the value

ln(|h^{∼}(η_{0}(1, π/3, 1/13))|) =

1, 5 ln(|(η_{0}(1, π/3, 1/13))^{2} − 1|^{2}) + 36 ln(13/12) − 6 ln 2+
18 ln(|η_{0}(1, π/3, 1/13)|^{2}) =

1, 5 ln(|(η_{0}(1, π/3, 1/13))^{2}−1|^{2})−6 ln(2)+18 ln((169/144)|η_{0}(1, π/3, 1/13)|^{2}). Further we have

ln(|(η_{0}(1, π/3, 1/13))^{2} − 1|^{2}) =
ln(17, 816127257...) + ln(32, 16071611...) = 6, 350849784...
1, 5 ln(|(η_{0}(1, π/3, 1/13))^{2} − 1|^{2}) = 9, 526274675...
ln((169/144)|η_{0}(1, π/3, 1/13)|^{2}) =
ln(169/144) + ln(23, 98842168...) = 3, 337656699... ,
18 ln((169/144)|η_{1}(1, π/3, 1/13)|^{2}) = 60, 07782059... ,
ln(64) = 4.158883083...

ln(|h^{∼}(η_{0}(1, π/3, 1/13))|) = 9, 526274675... − (61)

4, 158883083... + 60, 07782059... = 65, 44521218... .

L.A.Gutnik, On linear forms with coefficients in ℕζ(⊮ + ℕ), Part 7. 13

Let γ_{0}^{∗}(Δ) = ln(|h^{∼}(η_{0}(1, π/3, 1/Δ)|) + ω_{2}(Δ) (62) I view of (44) and (61), γ_{0}^{∗}(13) = 61.11644549... + 65, 44521218... = 126, 5616577..., (63)

ν(γ_{0}^{∗}(13) − ε) + O(1) ≤ (64)

ln(|h^{∼}(η_{0}(1, π/3, 1/13)|^{ν}(D^{∗∗}(ν))^{5}/(d_{1}^{∗}(ν)^{3}) ≤
ν(γ_{0}^{∗}(13) + ε) + O(1).

Let γ^{∗}(Δ) = γ_{0}^{∗}(Δ)/γ_{1}^{∗}(Δ). (65) In view of (60) and (64), there exist

− lim_{ν→+∞} \frac{ln(|h^{∼}(η_{0}(1, π/3, 1/13)|^{ν}(D^{∗∗}(ν))^{5}/(d_{1}^{∗}(ν)^{3})}{ln(|h^{∼}(η_{1}(1, π/3, 1/13)|^{ν}(D^{∗∗}(ν))^{5}/(d_{1}^{∗}(ν)^{3})} = (66)

γ^{∗}(13) (= 126, 5616577.../2, 911845783... = 43, 46440956...),

and γ^{∗}(13) < γ = 43, 464412 (67) (of course, if our Theorem is true for some γ, then all the bigger numbers can play the role of γ).

Let

h_{0}(ε) = \frac{γ_{0}^{∗}(13) + 6ε}{γ_{1}^{∗}(13) − 2ε}, (68)

where ε ∈ (0, γ_{1}^{∗}(13)/2). Clearly, h_{0}(ε) increases together with increasing ε in (0, γ_{1}^{∗}(13)/2), and, in view of (59), (65) and (67),

h_{0}(0) < γ.

Therefore there exists ε_{10} ∈ (0, γ_{1}^{∗}(13)/2) such that h_{0}(ε) < γ (69) for all the ε ∈ (0, ε_{10}). Let R_{1} = R_{1}(ε) = exp(γ_{1}^{∗}(13) + 2ε), (70)

R_{2} = R_{2}(ε) = exp(γ_{1}^{∗}(13) − 2ε), (71)

r_{1}^{∧} = r_{2}^{∧} = r^{∧} = r^{∧}(ε) = exp(γ_{0}^{∗}(13) + 2ε), (72)

L.A.Gutnik, On linear forms with coefficients in ℕζ(⊮ + ℕ), Part 7. 14

where ε ∈ (0, ε_{10}). Then R_{1} > R_{2} > 1, ln(R_{1}/R_{2}) = 4ε. (73) In view of (69), (ln(r^{∧}(ε)R_{1}(ε)/R_{2}(ε)))/ ln(R_{2}(ε)) = h_{0}(ε) < γ (74) for all the ε ∈ (0, ε_{10}). Let, as in (4.1.33) of the paper [65], α_{0}^{∗}(z; ν) = α^{∗}(z; ν), α_{1}^{∗}(z; ν) = β^{∗}(z; ν), α_{2}^{∗}(z; ν) = γ^{∗}(z; ν), (75)

α_{3}^{∗}(z; ν) = ϕ^{∗}(z; ν), α_{4}^{∗}(z; ν) = ψ^{∗}(z; ν), α_{5}^{∗}(z; ν) = ξ^{∗}(z; ν), (76) where ν ∈ ℕ. Let us consider the difference equation (29) in [70] now, i.e. the following difference equation:

∑_{j=0}^{4} q_{j}^{∧∗}(ν^{−1})y(ν + j) + y(ν + 5) = 0, (77)

where q_{j}^{∧∗}(ν^{−1}) ∈ ℚ(ν^{−1}), if j = 0 , …, 4, are well defined for ν ∈ [1/ε_{5}] + ℕ with some ε_{5} > 0 and q_{0}^{∧∗}(ν^{−1}) ≠ 0, when ν ∈ [1/ε_{5}] + ℕ. It is shown in [70] that the equation (77) has a solutions y(ν) = α_{k}^{∗}(1; ν), (78) where k = 0, 1, 3, 4, 5 and ν ∈ ℕ + [1/ε_{5}]; according to the Lemma 4.1.3 in [65], solutions (78) compose a basis of the space of the solutions of the equation (77). In view of (2.6) in (60),

α_{2}^{∗}(1, ν) = γ^{∗}(1, ν) = −(−1)^{ν}Res(R_{0}(t; ν))^{3}; t = ∞) = 0,

where ν ∈ ℕ. Therefore, in view of (2.21), (2.22), (2.19), (2.25) and (2.27), the five solutions y(ν) = α^{∗}(1; ν), y(ν) = β^{∗}(1; ν), (79)

y(ν) = f_{2}^{∗}(1; ν), y(ν) = f_{4}^{∗}(1; ν), y(ν) = f_{6}^{∗}(1; ν) (80) (like to the five solutions (78)) compose a basis of the space of the solutions of the equation (77). It is proved in [71], that equation (77) is the difference equation of Poincaré type and, if P (η) denotes the characterisic polynomial of this equation, then P (η) = F_{0}(η)(η − λ_{0}), (81) where λ_{0} ∈ ℚ,

F_{0}(η) = ∏_{l=0}^{1} ∏_{k=0}^{1} h^{∼}(η_{k,l}^{∗}) (82)

is the polynomial considered in the Lemma 3.1.6 of the of the paper [67], and

η_{k,l}^{∗} = η_{k}^{∧}(1, (2l − 1)π/3, 1/13) (83)

According to the Lemma 3.1.6 in [67], the polynomial F_{0}(η) is irreducible over ℚ; therefore the multiplicity of the root λ_{0} ∈ ℚ of the polynomial P (η) is equal to 1.

I apply further the following result [76]:

Lemma 7.6. Let s ∈ ℕ − 1, n ∈ ℕ,

a_{i}^{∼} ∈ ℂ, a_{i}(ν) ∈ ℂ,

a_{n}(ν) = 1, a_{i}(ν) − a_{i}^{∼} = O(1/ν) (84)

for ν ∈ ℕ and i = 0, …, n. Let us consider the following difference equation

∑_{k=0}^{n} a_{k}(ν)y(ν + k) = 0, (85)

where ν ∈ ℕ − 1. For m ∈ ℕ let V_{m} denotes the linear over ℂ space of solutions y = y(ν) of the equation

∑_{k=0}^{n} a_{k}(ν)y(ν + k) = 0, (86)

where ν ∈ m + ℕ − 1. Let the absolute values of all the roots of the characteristical polynomial

T (z) = ∑_{k=0}^{n} a_{k}^{∼}z^{k} (87)

are among the numbers {ρ_{i}: 1 ≤ i ≤ 1 + s}, such that ρ_{s+1} = 0 and ρ_{j} < ρ_{i} for 1 ≤ i < j ≤ s + 1. Let e_{i} and k_{i} denote respectively the sum and the maximum of the multiplicities of those roots, whose absolute value is equal to the number ρ_{i}, where i = 1, …, s + 1, and let k^{∗} = k_{s+1}. We suppose that, if s > 0, then

e_{i} > 0 (88)

for i = 1 …, s. For given y = y(ν) in ℂ^{m−1+ℕ}, let

ω_{n,y}(ν) = max(|y(ν)| , …, |y(ν + n − 1)|).

Then there exist A > 0, m ∈ ℕ, α^{∧}(ν) > 0 with ν ∈ m + ℕ − 1 and the subspaces V_{m,1}^{∨}, …, V_{m,s+1}^{∨} such that

lim_{ν→∞} α^{∧}(ν) = 0,

V_{m} = V_{m,1}^{∨} ⊕ …, ⊕ V_{m,s+1}^{∨}, dim_{ℂ}(V_{m,i}^{∨}) = e_{i}, 1 ≤ i ≤ s + 1,

and, if y ∈ V_{m,θ}^{∨} for some θ ∈ {1, ..., s}, then

exp(−A(ln(ν) + ν^{1−1/k_{θ}}))(ρ_{θ})^{ν}ω_{n}(y)(m) ≤ ω_{n,y}(ν) (89) for ν ∈ m + ℕ − 1; moreover, the spaces

V_{m,j}^{∧} = V_{m,j}^{∨} ⊕ … ⊕ V_{m,s+1}^{∨},

where j = 1 …, s + 1, and, if s ≥ 1 natural projections π_{j},

V_{m,j}^{∧} ↦ V_{m,j}^{∨},

where j = 1 …, s, have the folloving properties:

if y ∈ V_{m,θ}^{∧} for some θ ∈ {1, ..., s}, then

ω_{n,y}(ν) ≤ exp(A(ln(ν) + ν^{1−1/k_{θ}}))(ρ_{θ})^{ν}ω_{n,y}(m), (90)

(ω_{n,π_{θ(y)}}(m) − α(ν)ω_{n,y}(m))(ρ_{θ})^{ν} × (91)

exp(−A(ln(ν) + ν^{1−1/k_{θ}})) ≤ ω_{n,y}(ν),

where ν ∈ m + ℕ − 1; if

k^{∗} > 0, (92)

and y ∈ V_{m,s+1}^{∨} (= V_{m,s+1}^{∧}), then

|y(ν)| ≤ (A/ν)^{ν/k^{∗}}ω_{n,y}(m),

where ν ∈ m + ℕ − 1.

Proof. The proof is given in [76]. ■

Remark 7.1. Clearly, for any ε ∈ (0, 1) there exists C_{0} > 0 such that the inequalities (89) and (90) may be respectively replaced by the inequalities

(1/C_{0})(ρ_{θ} exp(−ε))^{ν}ω_{n}(y)(m) ≤ ω_{n,y}(ν) (93)

and

ω_{n,y}(ν) ≤ C_{0}(ρ_{θ} exp(ε))^{ν}ω_{n,y}(m). (94)

In our case the role of the equation (85) plays the equation (77) and the role of the polynomial (87) plays polynomial (81). In view (5.24) in [71], we have

n = 5, s = 3, ρ_{1} = |h^{∼}(η_{0}^{∧}(1, π/3, 1/13))| = |h^{∼}(η_{0}^{∧}(1, −π/3, 1/13))|,
ρ_{2} = |h^{∼}(η_{1}^{∧}(1, π, 1/13))| = h^{∼}(γ_{1})| = h(6/7),
ρ_{3} = |h^{∼}(η_{1}^{∧}(1, π/3, 1/13))| = |h^{∼}(η_{1}^{∧}(1, −π, 1/13))|.

Since the polynomial F_{0}(η) is irreducible over ℚ, it follows that

e_{1} = e_{3} = 2, e_{1} = k_{1} = k_{2} = k_{3} = 1, k_{4} = 0.

Since k_{4} = 0, it follows that V_{m,4}^{∨} = 0, V_{m,3}^{∧} = V_{m,3}^{∨}. In view (5.27) in [71], the space V_{m,3}^{∨} of the Lemma 7.6 contains f_{2+2k}^{∗}(1; ν) for k = 1, 2; moreover, since dim_{ℂ}(V_{m,3}^{∨}) = 2 and the five solutions (79) –(80), compose a basis of the space V_{m}, it follows that f_{4}^{∗}(1; ν), f_{6}^{∗}(1; ν) compose a basis of V_{m,3}^{∨} = V_{m,3}^{∧}. If

y = Z_{1}f_{4}^{∗}(1; ν) + Z_{2}f_{6}^{∗}(1; ν) (95)

L.A.Gutnik, On linear forms with coefficients in ℕζ(⊮ + ℕ), Part 7. 17

with {Z_{1}, Z_{2}} ⊂ ℂ, we let p^{∨}(y) = (max(|Z_{1}|, |Z_{2}|). (96) It is well known that function ω_{n,y}(m) (with variable y ∈ V_{m}) is a norm on the space V_{m}. We denote by p^{∧}(y) the restriction of the norm ω_{n,y}(m) on the space V_{m,3}^{∨} = V_{m,3}^{∧}. So, on the two-dimensional space V_{m,3}^{∨} = V_{m,3}^{∧} we have two norms p^{∨}(y) and p^{∧}(y). Therefore there exists a constant C_{1} > 0 such that p^{∨}(y) ≤ C_{1}p^{∧}(y), C_{1}p^{∨}(y) ≥ p^{∧}(y) (97) for any y ∈ V_{m,3}^{∨} = V_{m,3}^{∧}. In view of (93) – (94) and (97), (C_{0}C_{1}))^{−1}(ρ_{3} exp(−ε))^{ν}p^{∨}(y) ≤ ω_{n,y}(ν), (98)

ω_{n,y}(ν) ≤ C_{0}C_{1}(ρ_{3} exp(ε))^{ν}p^{∨}(y). (99) In view of (94), |α_{k}^{∗}(1; ν)| ≤ O(1)(ρ_{1} exp(ε))^{ν} (100) where k = 0, 1, 3, 4, 5 , ν ∈ ℕ and O(1) depends only from ε.

Let {m, n} ⊂ ℕ,

a_{i,k} ∈ ℝ

for i = 1 , …, m, k = 1 , …, n,

α_{j}^{∧}(ν) ∈ ℤ

where j = 1 , …, m + n and ν ∈ ℕ. Let there are γ_{0}^{∧}, r_{1}^{∧} ≥ 1, …, r_{m}^{∧} ≥ 1 such that |α_{i}(ν)| < γ_{0}^{∧}(r_{i}^{∧})^{ν} (101) where i = 1 , …, m and ν ∈ ℕ. Let

y_{k}(ν) = −α_{m+k}^{∧}(ν) + ∑_{i=1}^{m} a_{i,k}α_{i}^{∧}(ν)

where k = 1 , …, n and ν ∈ ℕ. If

X = (Z_{1} ; ⋮ ; Z_{n}) ∈ ℝ^{n}, (102)

then let q_{∞}(X) = max(|Z_{1}| , …, |Z_{n}|), (103)

y^{∧}(X) = y^{∧}(X, ν) = ∑_{k=1}^{n} y_{k}^{∧}(ν)Z_{k}

L.A.Gutnik, On linear forms with coefficients in ℕζ(⊮ + ℕ), Part 7. 18

for ν ∈ ℕ, let

ϕ_{i}(X) = ∑_{k=1}^{n} a_{i,k}Z_{k}

for i = 1 , …, m, and let

α_{0}^{∧}(X, ν) = ∑_{k=1}^{n} α_{m+k}^{∧}(ν)Z_{k}

for ν ∈ ℕ. Clearly,

y^{∧}(X, ν) = −α_{0}^{∧}(X, ν) + ∑_{i=1}^{m} α_{i}^{∧}(ν)ϕ_{i}(X)

for X ∈ ℜ^{n} and ν ∈ ℕ,

α_{0}^{∧}(X, ν) ∈ ℤ

for X ∈ ℤ^{n} and ν ∈ ℕ.

Lemma 7.7. Let \{l, n\} ⊂ ℕ, γ_{1}^{∧} > 0, γ_{2}^{∧} > \frac{1}{2}, R_{1} ≥ R_{2} > 1, α_{i} = (log(r_{i}^{∧}R_{1}/R_{2}))/ log(R_{2}), (104) where i = 1 , …, m, let X ∈ ℤ^{n}\{0},

γ_{3}^{∧} = γ_{1}^{∧}(R_{1})^{(− log(2γ_{2}R_{2}))/ log(R_{2})}, γ_{4}^{∧} = γ_{3}^{∧} (∑_{i=1}^{m} γ_{0}(r_{i}^{∧})^{(log(2γ_{2}^{∧}))/ log(R_{2})+l})^{−1}

and let for each ν ∈ ℕ − 1 hold the inequalities γ_{1}^{∧}(R_{1})^{−ν}q_{∞}(X) ≤ (105)

sup{|y^{∧}(X, κ)|: κ = ν, …, ν + l − 1},

|y^{∧}(X, ν)| ≤ γ_{2}^{∧}(R_{2})^{−ν}q_{∞}(X) (106) Then sup{∥ϕ_{i}(X)∥(q_{∞}^{(𝔎)}(X))^{α_{i}}: i = 1, …, m} ≥ γ_{4}^{∧}. (107) Proof. Proof may be found in [66], Theorem 2.3.1. ■

In our case

m = n = 2 , a_{1,1} = 2ζ(3), a_{1,2} = a_{2,1} = 3ζ(4), a_{2,2} = 6ζ(5),

α_{1}^{∧}(ν) = (D^{∗∗}(ν))^{5}/(d_{1}^{∗}(ν)^{3}))β^{∗}(1, ν),
α_{2}^{∧}(ν) = (D^{∗∗}(ν))^{5}/(d_{1}^{∗}(ν)^{3}))α^{∗}(1, ν),
α_{3}^{∧}(ν) = (D^{∗∗}(ν))^{5}/(d_{1}^{∗}(ν)^{3}))ψ^{∗}(1, ν),
α_{4}^{∧}(ν) = (D^{∗∗}(ν))^{5}/(d_{1}^{∗}(ν)^{3}))ξ^{∗}(1, ν).

According to the Lemma 7.5,

α_{j}^{∧}(ν) ∈ ℤ,

where j = 1 , …, 4 and ν ∈ ℕ. In view of (100) and (45),

|α_{j}^{∧}(ν)| ≤ (ρ_{1} exp(ε))^{ν} exp(ν(ω_{2}(13) + ε) + O(1)) = (108)

exp((γ_{0}^{∗}(13) + 2ε))ν + O(1),

where j = 1 , …, 4 , ν ∈ ℕ and O(1) depends only from ε.

So, there exists γ_{0} > 0 such that (101) holds with r_{j}^{∧} defined in (72).

The role of y_{k}(ν) of the Lemma 7.7 now play

y_{k}(ν) = (D^{∗∗}(ν))^{5}/(d_{1}^{∗}(ν)^{3}))f_{2+2k}^{∗}(1; ν),

where k = 1, 2, ν ∈ m + ℕ with m defined in the Lemma 8.1.

In view of (98), (99) and (45),

exp((γ_{3}^{∗}(13) − 2ε))ν + O(1))p^{∨}(y) = (109)

(ρ_{3} exp(−ε))^{ν} exp(ν(ω_{2}(13) − ε) + O(1))p^{∨}(y) ≤ ω_{2,y_{k}}(ν),

ω_{2,y_{k}}(ν) ≤ (110)

(ρ_{3} exp(ε))^{ν} exp(ν(ω_{2}(13) + ε) + O(1))p^{∨}(y) =
exp((γ_{3}^{∗}(13) + 2ε))ν + O(1))p^{∨}(y),

where O(1) depends only from ε, k = 1, 2, , ν ∈ m + ℕ with m defined in the Lemma 7.6; moreover in view of (95), (96) and (102),(103) with n = 2,

p^{∨}(y) = q_{∞}(X).

So, there exist γ_{1}^{∧} > 0, γ_{2}^{∧} > 0 such that the relations (105), (106) hold with R_{1}, R_{2} defined in (70) – (71), with l = n = 5 and with ν ∈ m + ℕ, where m is defined in the Lemma 7.6 (of course, the condition ν ∈ m + ℕ is not essential, because of the substitution ν := ν − m − 1 influences only on constants γ_{0}^{∧}, γ_{1}^{∧}, γ_{2}^{∧}, γ_{3}^{∧} and γ_{4}^{∧}). So, all the conditions of the lemma 7.7 are fullfilled. Therefore (107) holds, and, in view of (104) and (74),

α_{i} = h_{0}(ε) < γ,

where i = 1, 2 .■

§2. Verification of the previous calculations on computer and the values of γ^{∗}(Δ) for other Δ.

Let Z ∈ ℂ⧹(−∞, 0], Z = |Z|(cos(α) + i sin(α)) with |α| < π. Then

w(Z) = ((|Z| + Z)/2)/((|Z| + ℜ(Z))/2)^{1/2} =
(|Z| cos(α/2) + i sin(α/2))/(|Z|^{1/2})

and w(Z)^{2} = Z. Let

Z = X + iY = 1 − (1 − δ_{0}^{2})/(2z − 1)^{2} , ρ = ((|Z| + ℜ(Z))/2)^{1/2}.

L.A.Gutnik, On linear forms with coefficients in ℕζ(⊮ + ℕ), Part 7. 20

Then

|Z| = 2ρ^{2} − X , w = w(Z) = ρ + i\frac{Y}{2ρ}, ℜ(w(Z)) > 0. (111)

We put in (3.1.1) of the paper [62] η = λ(2z − 1)/(1 + δ_{0}), where z ≠ 1/2. Then, since z = −r exp(iψ), it follows that

D^{∨}(z, η) = (η + 1)(η + γ_{1}) + 2r exp(iψ)(1 + γ_{1})η =

η^{2} − 2(2z − 1)η/(1 + δ_{0}) + γ_{1} =
((2z − 1)/(1 + δ_{0}))^{2}(λ^{2} − 2λ + (1 − δ_{0}^{2})/(2z − 1)^{2}).

Clearly,

1 − (1 − δ_{0}^{2})/(2z − 1)^{2}) ∉ (−∞, 0]

if and only if

z ∉ (\frac{1}{2} − \frac{1}{2\sqrt{1 − δ_{0}^{2}}} , \frac{1}{2} + \frac{1}{2\sqrt{1 − δ_{0}^{2}}} ) ,

what we suppose further in conection with calculations of

|h^{∼}(η_{k}(1, π/3, δ_{0}))|

for k = 0, 1. Let W = w(1 − (1 − δ_{0}^{2})/(2z − 1)^{2}). Then

1/(2z − 1)^{2} = (1 − W^{2})/(1 − δ_{0}^{2}),

and

η_{k}^{∧}(r, ψ, δ_{0}) = \frac{2z − 1}{1 + δ_{0}}(1 + (−1)^{k}W) =

\frac{1 − δ_{0}}{2z − 1}(1 − (−1)^{k}W)^{−1},

η_{k}^{∧}(r, ψ, δ_{0})^{2} = \frac{(2z − 1)^{2}}{(1 + δ_{0})^{2}} (1 + (−1)^{k}W)^{2} =

\frac{(1 − δ_{0})^{2}}{(2z − 1)^{2}} (1 − (−1)^{k}W)^{−2} =

\frac{1 − δ_{0}}{1 + δ_{0}} \frac{1 + (−1)^{k}W}{1 − (−1)^{k}W} =

\frac{1 − (−1)^{k}δ_{0}W + (−1)^{k}W − δ_{0}}{1 − (−1)^{k}δ_{0}W − (−1)^{k}W + δ_{0}} ,

1 − η_{k}^{∧}(r, ψ, δ_{0})^{2} = 2\frac{δ_{0} − (−1)^{k}W}{(1 + δ)(1 − (−1)^{k}W)} =

2(2z − 1)^{2}(1 + (−1)^{k}W)\frac{δ_{0} − (−1)^{k}W}{(1 + δ)^{2}(1 − δ_{0})}.

Let

A_{k}|1 + (−1)^{k}W |^{2} = 1 + |Z| + 2(−1)^{k}ℜ(W) =
1 + |Z| + 2(−1)^{k}ρ = (1 + (−1)^{k}ρ)^{2} + ρ^{2} − X,

B_{k} = |δ_{0} − (−1)^{k}W |^{2} = δ_{0}^{2} + |Z| − 2(−1)^{k}δ_{0}ℜ(W) =
δ_{0}^{2} + |Z| − 2(−1)^{k}δ_{0}ρ = (δ_{0} − (−1)^{k}ρ)^{2} + ρ^{2} − X,

where k = 0, 1. Then

|h^{∼}(η_{k}(1, π/3, 1/Δ))| = 2^{−6}(1 − δ_{0})^{3−3Δ}×

(\frac{|2z − 1|^{2}}{(1 + δ_{0})^{2}} A_{k})^{(3Δ−3)/2} ×

2^{3}|(2z − 1)^{2}|^{3}A_{k}^{3/2} \frac{B_{k}^{3/2}}{(1 + δ)^{6}(1 − δ_{0})^{3}} =

2^{−3}(1 − δ_{0}^{2})^{−3Δ}(|2z − 1|^{2})^{(3Δ+3)/2}(1 + δ_{0})^{−3}A_{k}^{3Δ/2}B_{k}^{3/2}.

In our case

z = − exp(πi/3), 2z − 1 = −2 − i\sqrt{3}, |2z − 1|^{2} = 7,

Z = X + iY = 1 − \frac{Δ^{2} − 1}{49Δ^{2}}(1 − 4i\sqrt{3}) =

\frac{48Δ^{2} + 1}{49Δ^{2}} + i\frac{4Δ^{2} − 4}{49Δ^{2}} \sqrt{3},

|Z| = (48Δ^{4} + 1)^{1/2}/(7Δ^{2}),

X = \frac{48Δ^{2} + 1}{49Δ^{2}}, ρ = ((|Z| + X)/2)^{1/2} =

(7(48Δ^{4} + 1)^{1/2} + 48Δ^{2} + 1)/98)^{1/2}/Δ.

Let

ω_{k}(Δ) = ln |h^{∼}(η_{k}(1, π/3, 1/Δ))|) =
−3 ln(2) + \frac{3Δ + 3}{2} ln(7) + 3Δ ln(Δ^{2}/(Δ^{2} − 1))−
3Δ ln((Δ + 1)/Δ) + \frac{3Δ}{2} ln(A_{k}) + \frac{3}{2} ln(B_{k}),

where k = 0, 1. In view of (58), (62) and (65),

γ_{k}^{∗}(Δ) = (−1)^{k}ω_{k}(Δ) + (−1)^{k}ω_{2}(Δ),
γ^{∗}(Δ) = (ω_{0}(Δ) + ω_{2}(Δ)/(ω_{1}(Δ) − ω_{2}(Δ)).

The following program (written in Turbo Pascal) was used in [58] for calculations:

Program Fz5hh;

label 1,2;

var k,b:shortint;Delta:real {Delta = Δ};

absZ,rho,A0,A1,B0,B1,om2,omh,omhh,om1,om0,s0,s1,mu:real {absZ = |Z|, rho = ρ, A0 = A_{0}, A1 = A_{1}, } {B0 = B_{0}, B1 = B_{1}, om2 = ω_{2}, om1 = ω_{2}, om0 = ω_{0}}

{s0 = ∑_{1≤κ≤(Δ−1)/2} cot(πκ/(Δ − 1))}

{s1 = ∑_{1≤κ≤(Δ+1)/2} cot(πκ/(Δ + 1))}

{mu = γ^{∗}(Δ)}; begin; 1: writeln(’continue?,yes=1,no=0’); readln (b); if (b = 0) then goto 2; writeln(’Delta=?’); readln(Delta); absZ := sqrt(48 ∗ sqr(sqr(Delta)) + 1)/(7 ∗ sqr(Delta)); rho := sqrt((7 ∗ sqrt(48 ∗ sqr(sqr(Delta)) + 1)+ 48 ∗ sqr(Delta) + 1)/98)/Delta; A0 := 1 + absZ + 2 ∗ rho; A1 := 1 + absZ − 2 ∗ rho; B0 := 1/sqr(Delta) + absZ − 2 ∗ rho/Delta; B1 := 1/sqr(Delta) + absZ + 2 ∗ rho/Delta; s0 := 0; k := 1; while (k < (Delta − 1)/2) do begin s0 := s0 + cos(pi ∗ k/(Delta − 1))/ sin(pi ∗ k/(Delta − 1)); k := k + 1; end; s1 := 0; k := 1; while (k < (Delta + 1)/2) do begin s1 := s1 + cos(pi ∗ k/(Delta + 1))/ sin(pi ∗ k/(Delta + 1)); k := k + 1; end; om2 := 5 ∗ (Delta + 1) − (3/2) ∗ (Delta − 1) ∗ ln(Delta/(Delta − 1))+ (3/2) ∗ (Delta + 1) ∗ ln((Delta + 1)/Delta) + 3 ∗ pi ∗ s0/2 − 3 ∗ pi ∗ s1/2; om1 := −3 ∗ ln(2) + ((3 ∗ Delta + 3)/2) ∗ ln(7)+ 3 ∗ Delta ∗ ln(sqr(Delta)/(sqr(Delta) − 1))− 3 ∗ ln((Delta + 1)/Delta) + (3 ∗ Delta/2) ∗ ln(A1) + (3/2) ∗ ln(B1); om0 := −3 ∗ ln(2) + ((3 ∗ Delta + 3)/2) ∗ ln(7)+ 3 ∗ Delta ∗ ln(sqr(Delta)/(sqr(Delta) − 1))− 3 ∗ ln((Delta + 1)/Delta) + (3 ∗ Delta/2) ∗ ln(A0) + (3/2) ∗ ln(B0); mu := −(om0 + om2)/(om1 + om2); writeln(’Delta=’,Delta,’,’,’mu=’,mu); goto 1; 2: end. (It is not necessary to know Turbo Pascal for the use of this program; it

is sufficient to rewrite it in MS-DOS format (not in TeX), to place it in the directory of Turbo Pascal, give a command (in command line) turbo fz5hh and then hit Ctrl+F9).

In this program mu = γ^{∗}(Δ). The following results are obtained:

Δ = 2, mu = −5, 69729529498727... ,

Δ = 3, mu = −18, 4506653923600... ,

Δ = 4, mu = −42, 0868798934971... ,

Δ = 5, mu = −3766, 31419552490... ,

Δ = 6, mu = 128, 630462898873... ,

Δ = 7, mu = 65, 78503337027339... ,

Δ = 8, mu = 56, 0423180866055... ,

Δ = 9, mu = 48, 4350925860344... ,

Δ = 10, mu = 46, 5570123894140... ,

Δ = 11, mu = 44, 3334143236862... ,

Δ = 12, mu = 44, 1044453638024...,

Δ = 13, mu = 43, 46441140066032... ,

Δ = 14, mu = 43, 8257111129351... ,

Δ = 15, mu = 43, 8522467170842... ,

Δ = 16, mu = 44, 4821306835511... ,

Δ = 17, mu = 44, 8464590454823... ,

Δ = 18, mu = 45, 6162272955407... ,

Δ = 19, mu = 46, 61723601535778... ,

Δ = 20, mu = 47, 0213561055716... .

We see that Δ = 6 is the first Δ with γ^{∗}(Δ) > 0.

§3. Epilogue.

Much water has flown under the bridges since I announced the above results for the first time. During those eight years, new publications in this subject have appeared. First of all, I wish to refer to the deep research by Rhin and Viola [33]; let me also cite the very interesting papers by Hata [34], by Boll and Rivoal [36], and by Zudilin [37], [38].

References.

[1] R.Apéry, Interpolation des fractions continues et irrationalite de certaines constantes, Bulletin de la section des sciences du C.T.H., 1981, No 3, 37 – 53;

[2] F.Beukers, A note on the irrationality of ζ(2) and ζ(3), Bull. London Math. Soc., 1979, 11, 268 – 272;

[3] A.van der Porten, A proof that Euler missed...Apéry’s proof of the irrationality of ζ(3), Math Intellegencer, 1979, 1, 195 – 203;

[4] W. Maier, Potenzreihen irrationalen Grenzwertes, J.reine angew. Math., 156, 1927, 93 – 148;

[5] E.M. Nikis̆in, On irrationality of the values of the functions F(x,s) (in Russian), Mat.Sb. 109 (1979), 410 – 417; English transl. in Math. USSR Sb. 37 (1980), 381 – 388;

[6] G.V. Chudnovsky, Pade approximations to the generalized hyper-geometric functions I,J.Math.Pures Appl., 58, 1979, 445 – 476; [7] , Transcendental numbers, Number Theory,Carbondale,

Lecture Notes in Math, Springer-Verlag, 1979, 751, 45 – 69;

[8] , Approximations rationelles des logarithmes de nombres rationelles C.R.Acad.Sc. Paris, Série A, 1979, 288, 607 – 609;

[9] , Formules d’Hermite pour les approximants de Padé de logarithmes et de fonctions binômes, et mesures d’irrationalité, C.R.Acad.Sc. Paris, Série A, 1979, t.288, 965 – 967;

[10] ,Un systḿe explicite d’approximants de Padé pour les fonctions hypérgéometriques généraliesées, avec applications a l’arithmétique, C.R.Acad.Sc. Paris, Série A, 1979, t.288, 1001 – 1004;

[11] , Recurrenses defining Rational Approximations to the irrational numbers, Proceedings of the Japan Academie, Ser. A, 1982, 58, 129 – 133;

[12] , On the method of Thue-Siegel, Annals of Mathematics, 117 (1983), 325 – 382;

[13] K.Alladi and M. Robinson, Legendre polinomials and irrationality, J. Reine Angew.Math., 1980, 318, 137 – 155;

[14] A. Dubitskas, An approximation of logarithms of some numbers, Diophantine approximations II,Moscow, 1986, 20 – 34;

[15] , On approximation of π/\sqrt{3} by rational fractions, Vestnik MGU, series 1, 1987, 6, 73 – 76;

[16] S.Eckmann, Über die lineare Unaqbhangigkeit der Werte gewisser Reihen, Results in Mathematics, 11, 1987, 7 – 43;

[17] M.Hata, Legendre type polinomials and irrationality mesures, J. Reine Angew. Math., 1990, 407, 99 – 125;

[18] A.O. Gelfond, Transcendental and algebraic numbers (in Russian), GIT-TL, Moscow, 1952;

[19] H.Bateman and A.Erdélyi, Higher transcendental functions,1953, New-York – Toronto – London, Mc. Grow-Hill Book Company, Inc.;

[20] E.C.Titchmarsh, The Theory of Functions, 1939, Oxford University Press;

[21] E.T.Whittaker and G.N. Watson, A course of modern analysis, 1927, Cambridge University Press;

[22] O.Perron, Über die Poincaresche Differenzengleichumg, Journal für die reine und angewandte mathematik, 1910, 137, 6 – 64;

[23] A.O.Gelfond, Differenzenrechnung (in Russian), 1967, Nauka, Moscow.

[24] A.O.Gelfond and I.M.Kubenskaya, On the theorem of Perron

in the theory of differrence equations (in Russian),

IAN USSR, math. ser., 1953, 17, 2, 83 – 86.

[25] M.A.Evgrafov, New proof of the theorem of Perron (in Russian),IAN USSR, math. ser., 1953, 17, 2, 77 – 82;

[26] G.A.Frejman, On theorems of of Poincaré and Perron (in Russian), UMN, 1957, 12, 3 (75), 243 – 245;

[27] N.E.Nörlund, Differenzenrechnung, Berlin, Springer Verlag, 1924;

[28] I.M.Vinogradov, Foundtions of the Number Theory, (in Russian), 1952, GIT-TL;

[29] , J. Diedonne, Foundations of modern analysis, Institut des Hautes Études Scientifiques, Paris, Academic Press, New York and London, 1960

[30] CH.-J. de la Vallée Poussin, Course d’analyse infinitésimale, Russian translation by G.M.Fikhtengolts, GT-TI, 1933;

[31] H.WEYL, Algebraic theory of numbers, 1940, Russian translation by L.I.Kopejkina;

[32] G.Rhin and C. Viola. On a prmutation group related to ζ(2), Acta Arithmetica 77(1996), 25 – 56;

[33] G.Rhin and C. Viola. The group structure for ζ(3), Acta Arithmetica 97(2001), 269 – 293; [34] M. Hata. A new irrationality measure forζ(3),

Acta Arithmetica 92(2000), 47 – 57;

[35] T.Rivual. La focntion zêta de Riemann prend une infinité valeurs irrationnelles aux entires impairs, C.R.Acad.Sci.Paris, série 1, p. 267 – 270, 2000.

[36] K.Boll, T.Rivual. Irrationalité d‘une infinité valeurs la focntion zêta aux entires impairs, Invent. math., 146, 193 – 207 (2001);

[37] W.Zudilin. One from the nubers ζ(5), ζ(7), ζ(9), ζ(11) is irrational.

(in Russian) UMN 56(4) (2001), 149 – 150;

[38] W.Zudilin. Apéry’s Theorem and problems for the values of ζ-function of Riemann and their q−analogs. (in Russian) Moscow state university, doctoral thesis, Moscow 2004;

[38] L.A.Gutnik, On the decomposition of the difference operators of Poincaré type (in Russian), VINITI, Moscow, 1992, 2468 – 92, 1 – 55;

[39] , On the decomposition of the difference operators of Poincaré type in Banach algebras (in Russian), VINITI, Moscow, 1992, 3443 – 92, 1 – 36;

[40] , On the difference equations of Poincaré type (in Russian), VINITI, Moscow 1993, 443 – B93, 1 – 41;

[41] , On the difference equations of Poincaré type in normed algebras (in Russian), VINITI, Moscow, 1994, 668 – B94, 1 – 44;

[42] , On the decomposition of the difference equations of Poincaré type (in Russian), VINITI, Moscow, 1997, 2062 – B97, 1 – 41;

[43] , The difference equations of Poincaré type with characteristic polynomial having roots equal to zero (in Russian), VINITI, Moscow, 1997, 2418 – 97, 1 – 20;

[44] , On the behavior of solutions of difference equations of Poincaré type (in Russian), VINITI, Moscow, 1997, 3384 – B97, 1 - 41;

[45] , On the variability of solutions of difference equations of Poincaré type (in Russian), VINITI, Moscow, 1999, 361 – B99, 1 – 9;

[46] , To the question of the variability of solutions of difference equations of Poincaré type (in Russian), VINITI, Moscow, 2000, 2416 – B00, 1 – 22;

[47] , On linear forms with coefficients in ℕζ(1 + ℕ), Max-Plank-Institut für Mathematik, Bonn, Preprint Series, 2000, 3, 1 – 13;

[48] , On the Irrationality of Some Quantyties Containing ζ(3) (in Russian), Uspekhi Mat. Nauk, 1979, 34, 3(207), 190;

[49] , On the Irrationality of Some Quantities Containing ζ(3), Eleven papers translated from the Russian, American Mathematical Society, 1988, 140, 45 - 56;

[50] , Linear independence over ℚ of dilogarithms at rational points (in Russian), UMN, 37 (1982), 179-180; english transl. in Russ. Math. surveys 37 (1982), 176-177;

[51] , On a measure of the irrationality of dilogarithms at rational points (in Russian), VINITI, 1984, 4345-84, 1 – 74;

[52] , To the question of the smallness of some linear forms (in Russian), VINITI, 1993, 2413-B93, 1 – 94;

[53] , About linear forms, whose coefficients are logarithms of algebraic numbers (in Russian), VINITI, 1995, 135-B95, 1 – 149;

[54] , About systems of vectors, whose coordinates are linear combinations of logarithms of algebraic numbers with algebraic coefficients (in Russian), VINITI, 1994, 3122-B94, 1 – 158;

[55] , On the linear forms, whose coefficients are 𝔸 - linear combinations of logarithms of 𝔸 - numbers, VINITI, 1996, 1617-B96, pp. 1 – 23.

[56] , On systems of linear forms, whose coefficients are 𝔸 - linear combinations of logarithms of 𝔸 - numbers, VINITI, 1996, 2663-B96, pp. 1 – 18.

[57] , About linear forms, whose coefficients are ℚ-proportional to the number log 2, and the values of ζ(s) for integer s (in Russian), VINITI, 1996, 3258-B96, 1 – 70;

[58] , The lower estimate for some linear forms, coefficients of which are proportional to the values of ζ(s) for integer s (in Russian), VINITI, 1997, 3072-B97, 1 – 77;

[59] , On linear forms with coefficients in ℕζ(1 + ℕ)

Max-Plank-Institut für Mathematik, Bonn, Preprint Series, 2000, 3, 1 – 13;

[60] , On linear forms with coefficients in ℕζ(1 + ℕ)

(the detailed version,part 1), Max-Plank-Institut für Mathematik, Bonn, Preprint Series, 2001, 15, 1 – 20;

[61] , On linear forms with coefficients in ℕζ(1 + ℕ)

(the detailed version,part 2), Max-Plank-Institut für Mathematik, Bonn, Preprint Series,2001, 104, 1 – 36;

[62] , On linear forms with coefficients in ℕζ(1 + ℕ)

(the detailed version,part 3), Max-Plank-Institut für Mathematik, Bonn, Preprint Series, 2002, 57, 1 – 33;

[63] , On the rank over ℚ of some real matrices (in Russian), VINITI, 1984, 5736-84; 1 – 29;

[64] , On the rank over ℚ of some real matrices, Max-Plank-Institut für Mathematik, Bonn, Preprint Series, 2002, 27, 1 – 32;

[65] , On linear forms with coefficients in ℕζ(1 + ℕ)

(the detailed version, part 4), Max-Plank-Institut für Mathematik, Bonn, Preprint Series, 2002, 142, 1 – 27;

[66] , On the dimension of some linear spaces over finite extension of ℚ (part 2), Max-Plank-Institut für Mathematik, Bonn, Preprint Series, 2002, 107, 1 – 37;

[67] , On the dimension of some linear spaces over ℚ (part 3), Max-Plank-Institut für Mathematik, Bonn, Preprint Series, 2003, 16, 1 – 45.

[68] ,On the difference equation of Poincaré type (Part 1).

Max-Plank-Institut für Mathematik, Bonn, Preprint Series, 2003, 52, 1 – 44.

[69] , On the dimension of some linear spaces over ℚ, (part 4)

Max-Plank-Institut für Mathematik, Bonn, Preprint Series, 2003, 73, 1 – 38.

[70] , On linear forms with coefficients in ℕζ(1 + ℕ)

(the detailed version, part 5), Max-Plank-Institut für Mathematik, Bonn, Preprint Series, 2003, 83, 1 – 13.

[71] , On linear forms with coefficients in ℕζ(1 + ℕ)

(the detailed version, part 6), Max-Plank-Institut für Mathematik, Bonn, Preprint Series, 2003, 99, 1 – 33.

[72] ,On the difference equation of Poincaré type (Part 2).

Max-Plank-Institut für Mathematik, Bonn, Preprint Series, 2003, 107, 1 – 25.

[73] , On the asymptotic behavior of solutions of difference equation (in English). Chebyshevskij sbornik, 2003, v.4, issue 2, 142 – 153.

[74] , On linear forms with coefficients in ℕζ(1 + ℕ), Bonner Mathematishe Schriften Nr. 360, Bonn, 2003, 360.

[75] , On linear combinations of logarithms of algebraic numbers with algebraic coefficients. International conference ”Diophantine analysis, uniform distributions and applications” August 25-30, 2003, Minsk, Belarus Abstracts pp. 16 – 17.

[76] ,On the difference equation of Poincaré type (Part 3).

Max-Plank-Institut für Mathematik, Bonn, Preprint Series, 2004, 09, 1 – 34.

[77] , On the dimension of some linear spaces over ℚ, (part 5)

Max-Plank-Institut für Mathematik, Bonn, Preprint Series, 2004, 46, 1 – 42.

E-mail: gutnik@gutnik.mccme.ru