Toward verification of the Riemann hypothesis: Application of the Li criterion

TL;DR

This paper explores the application of the Li criterion to verify the Riemann hypothesis by analyzing a sequence derived from the Riemann xi function and proposing conjectures that, if true, confirm the hypothesis.

Contribution

It introduces new bounds for zeta function sums and the sequence , and links the Riemann hypothesis to properties of the  sequence and conjectured characteristics.

Findings

Derived new bounds for zeta sums and  sequence

Proposed conjecture linking  properties to the Riemann hypothesis

Established that the hypothesis would follow if the conjecture holds

Abstract

We substantially apply the Li criterion for the Riemann hypothesis to hold. Based upon a series representation for the sequence \{\lambda_k\}, which are certain logarithmic derivatives of the Riemann xi function evaluated at unity, we determine new bounds for relevant Riemann zeta function sums and the sequence itself. We find that the Riemann hypothesis holds if certain conjectured properties of a sequence \eta_j are valid. The constants \eta_j enter the Laurent expansion of the logarithmic derivative of the zeta function about s=1 and appear to have remarkable characteristics. {\em On our conjecture}, not only does the Riemann hypothesis follow, but an inequality governing the values \lambda_n and inequalities for the sums of reciprocal powers of the nontrivial zeros of the zeta function.