Densities, Laplace Transforms and Analytic Number Theory

TL;DR

This paper explores the connections between densities, Laplace transforms, and analytic number theory, providing new derivations and interpretations related to the Riemann Hypothesis and Li's criterion.

Contribution

It offers a novel derivation of the explicit formula using Laplace transforms and presents an alternative probabilistic interpretation of Li's criterion.

Findings

Derived an explicit formula based on Laplace transforms.

Provided an alternative expression for Li's criterion.

Suggested a probabilistic perspective on the Riemann Hypothesis.

Abstract

Li showed that the Riemann Hypothesis is equivalent to the nonnegativity of a certain sequence of numbers. Bombieri and Lagarias gave an arithmetic formula for the number sequence based on the Guinand-Weil explicit formula and showed that Li's criterion is equivalent to Weil's criterion for the Riemann Hypothesis. We provide a derivation of the explicit formula based on Laplace transforms and present an alternative expression for Li's criterion that invites a probabilistic interpretation.