Irrationality of the sums of zeta values

Pilehrood, Tatiana Hessami; Pilehrood, Kh. Hessami

doi:10.1007/s11006-006-0063-1

Cited by 4 publications

(3 citation statements)

References 7 publications

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“…The integers s k,i depend also implicitly on n, a, r, N, f and p. Their values for i ≡ p mod 2 do not appear in the linear combinations of part (iii), but they could be of interest in other settings. Another feature of this construction is that for i ≤ a, the integers s k,i depend only on n, a, r, N but not on f or p. Probably this could lead to variants of our results in the style of [14] or [7].…”

Section: Statement Of the Resultsmentioning

confidence: 99%

Irrationality of values of L‐functions of Dirichlet characters

Fischler

2019

J. London Math. Soc.

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In a recent paper with Sprang and Zudilin, the following result was proved: if a is large enough in terms of ε > 0, then at least 2 (1−ε) log a log log a values of the Riemann zeta function at odd integers between 3 and a are irrational. This improves on the Ball-Rivoal theorem, that provides only 1−ε 1+log 2 log a such irrational values -but with a stronger property: they are linearly independent over the rationals.In the present paper we generalize this recent result to both L-functions of Dirichlet characters and Hurwitz zeta function. The strategy is different and less elementary: the construction is related to a Padé approximation problem, and a generalization of Shidlovsky's lemma is used to apply Siegel's linear independence criterion.We also improve the analogue of the Ball-Rivoal theorem in this setting: we obtain 1−ε 1+log 2 log a linearly independent values L(f, s) with s ≤ a of a fixed parity, when f is a Dirichlet character. The new point here is that the constant 1 + log 2 does not depend on f .

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Section: Statement Of the Resultsmentioning

confidence: 99%

Irrationality of values of L‐functions of Dirichlet characters

Fischler

2019

J. London Math. Soc.

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“…To conclude this introduction we mention the following result, analogous to the one of [8] concerning the numbers λ 0 ζ(s) + λ 1 sζ(s + 1) (see also Théorème 2 of [4]). Theorem 1.7.…”

Section: Introductionmentioning

confidence: 97%

Distribution of irrational zeta values

Fischler¹

2017

Bul. Soc. Math. France

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In this paper we refine Ball-Rivoal's theorem by proving that for any odd integer a sufficiently large in terms of ε > 0, there exist [ (1−ε) log a 1+log 2 ] odd integers s between 3 and a, with distance at least a ε from one another, at which Riemann zeta function takes Q-linearly independent values. As a consequence, if there are very few integers s such that ζ(s) is irrational, then they are rather evenly distributed. The proof involves series of hypergeometric type estimated by the saddle point method, and the generalization to vectors of Nesterenko's linear independence criterion.

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“…To conclude this introduction we mention the following result, analogous to the one of [8] concerning the numbers λ 0 ζ(s) + λ 1 sζ(s + 1) (see also Théorème 2 of [4]).…”

Section: Introductionmentioning

confidence: 99%

Distribution of irrational zeta values

Fischler¹

2013

Preprint

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In this paper we refine Ball-Rivoal's theorem by proving that for any odd integer a sufficiently large in terms of ε > 0, there exist [ (1−ε) log a 1+log 2 ] odd integers s between 3 and a, with distance at least a ε from one another, at which Riemann zeta function takes Q-linearly independent values. As a consequence, if there are very few integers s such that ζ(s) is irrational, then they are rather evenly distributed.The proof involves series of hypergeometric type estimated by the saddle point method, and the generalization to vectors of Nesterenko's linear independence criterion.

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Irrationality of the sums of zeta values

Cited by 4 publications

References 7 publications

Irrationality of values of L‐functions of Dirichlet characters

Irrationality of values of L‐functions of Dirichlet characters

Distribution of irrational zeta values

Distribution of irrational zeta values

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