From asymptotic to closed forms for the Keiper/Li approach to the Riemann Hypothesis

TL;DR

This paper derives explicit closed-form variants of the Keiper--Li sequence related to the Riemann Hypothesis, enabling easier computation and potential detection of zeros violating RH.

Contribution

It introduces a new sequence in elementary closed form that retains the sensitivity to RH and simplifies analysis compared to traditional Keiper--Li constants.

Findings

New sequence signals RH violations through its behavior.

Closed-form expressions facilitate faster computations.

Analysis of Davenport--Heilbronn counterexamples demonstrates the sequence's effectiveness.

Abstract

The Riemann Hypothesis (RH) - that all nonreal zeros of Riemann's zeta function shall have real part 1/2 - remains a major open problem. Its most concrete equivalent is that an infinite sequence of real numbers, the Keiper--Li constants, shall be everywhere positive (Li's criterion). But those numbers are analytically elusive and strenuous to compute, hence we seek simpler variants. The essential sensitivity to RH of that sequence lies in its asymptotic tail; then, retaining this feature, we can modify the Keiper--Li scheme to obtain a new sequence in elementary closed form. This makes for a more explicit analysis, with easier and faster computations. We can moreover show how the new sequence will signal RH-violating zeros if any, by observing its analogs for the Davenport--Heilbronn counterexamples to RH.