A note on the number of irrational odd zeta values. (9th August 2020)

• Record Type: Journal Article
• Authors: Lai, Li; Yu, Pin
• Is Part Of: Compositio mathematica. Volume 156:Number 8(2020)
• Page Start: 1699
• Page End: 1717
• Publication Date: 2020-08-09
• Subjects: irrationality, -- zeta values, -- hypergeometric series; 11J72, -- 11M06, -- 33C20; Mathematics -- Periodicals; 510
• DOI: 10.1112/S0010437X20007307
• Languages: English
• ISSNs: 0010-437X

Abstract

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.