Analytic Extension of Keiper-Li Coefficients

Krzysztof Maślanka

Published 2022-11-14Version 1

We construct certain entire function $\lambda (s)$ which for integer $s$ coincides with the well-known Keiper-Li coefficients, i.e. $\lambda (n)=\lambda_{n}$. This is an even function $% \lambda (s)=\lambda (-s)$ and has an infinitude of complex zeros exhibiting interesting distribution. Extensive tables of more than $3500$ complex zeros of $\lambda (s)$ with precision of $14$ significant digits are included. A detailed analysis of the distribution of these zeros may shed some light on the Riemann hypothesis. It turns out that possible violation of the Riemann hypothesis (if such is the case) would be clearly reflected in the specific distribution of these zeros. Key words: Riemann hypothesis, Keiper-Li coefficients, interpolation in unequally spaced nodes.