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A note on the number of irrational odd zeta values

Part of: Diophantine approximation, transcendental number theory Zeta and $L$-functions: analytic theory

Published online by Cambridge University Press:  09 October 2020

Li Lai  and
Pin Yu
Li Lai
Affiliation:
Department of Mathematics and Yau Mathematical Sciences Center, Tsinghua University, 100084 Beijing, China [email protected]
Pin Yu
Affiliation:
Department of Mathematics and Yau Mathematical Sciences Center, Tsinghua University, 100084 Beijing, China [email protected]
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Abstract

We prove that, for any small $\varepsilon > 0$, the number of irrationals among the following odd zeta values: $\zeta (3),\zeta (5),\zeta (7),\ldots ,\zeta (s)$ is at least $( c_0 - \varepsilon )({s^{1/2}}/{(\log s)^{1/2}})$, provided $s$ is a sufficiently large odd integer with respect to $\varepsilon$. The constant $c_0 = 1.192507\ldots$ can be expressed in closed form. Our work improves the lower bound $2^{(1-\varepsilon )({\log s}/{\log \log s})}$ of the previous work of Fischler, Sprang and Zudilin. We follow the same strategy of Fischler, Sprang and Zudilin. The main new ingredient is an asymptotically optimal design for the zeros of the auxiliary rational functions, which relates to the inverse totient problem.

Keywords

irrationalityzeta valueshypergeometric series

MSC classification

Primary: 11J72: Irrationality; linear independence over a field
Secondary: 11M06: $zeta (s)$ and $L(s, chi)$ 33C20: Generalized hypergeometric series, $_pF_q$
Type
Research Article
Information
Compositio Mathematica , Volume 156 , Issue 8 , August 2020 , pp. 1699 - 1717
Copyright
Copyright © The Author(s) 2020

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