Densities, Laplace Transforms and Analytic Number Theory

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.