Infinite set of non linear Equations for the Li- Keiper Coefficients: a possible new upper and lower bound

TL;DR

This paper introduces new bounds for Li-Keiper coefficients derived from an infinite set of nonlinear equations, supported by numerical experiments up to n=15 and 32, with implications for understanding the Riemann zeta function.

Contribution

It proposes a novel approach to bounding Li-Keiper coefficients using nonlinear equations and cluster functions, offering potential insights into the Riemann Hypothesis.

Findings

Proposed new upper and lower bounds for Li-Keiper coefficients.

Numerical validation of bounds up to n=15.

Counting zeros related to Glaisher-Kinkelin constant up to n=32.

Abstract

Starting with an infinite set of non linear Equations for the Li-Keiper coefficients, we first specify a lower bound emerging from the infinite set and give a characterization of it. Then, we propose a possible new upper and lower bound for the coefficients in few of the partitions occurring in the cluster functions furnishing in a nonlinear way the coefficients. A numerical experiment up to n=15 confirms the proposed bounds and an experiment, i.e. the counting of the zeros in the binary representation of an integer for a constant related to the Glaisher-Kinkelin constant is also given up to n=32.