Analytic Extension of Keiper-Li Coefficients

TL;DR

This paper constructs an entire function extending Keiper-Li coefficients, analyzes its complex zeros, and discusses implications for the Riemann hypothesis based on zero distribution patterns.

Contribution

It introduces an analytic extension of Keiper-Li coefficients and provides extensive zero distribution data to explore implications for the Riemann hypothesis.

Findings

Zeros form quadruplets if RH is true

Zeros align as doublets if RH is false

Extensive zero data with high precision included

Abstract

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. More specifically, they form complex quadruplets if the Riemann hypothesis is true, and aligned real doublets if it is false.