On the Riemann zeta-function, Part II

TL;DR

This paper constructs a meromorphic function from the Riemann zeta-function, determines its Laplace transform representation in specific regions, and explores implications related to the Riemann hypothesis without relying on unproven assumptions.

Contribution

It provides a novel analysis of a function derived from the zeta-function, including Laplace transform representations and positivity properties, independent of unproven hypotheses.

Findings

Laplace transform representation of f(s) on specific vertical strips

Positivity of the Laplace density for certain parameters

Conditional results on the zeta-function assuming hypotheses

Abstract

An odd meromorphic function f(s) is constructed from the Riemann zeta-function evaluated at one-half plus s. We determine the two-sided Laplace transform representation of f(s) on open vertical strips, V'(4w), disjoint from the (translated) critical strip. V'(4w) consists of all s with real part, Re(s), of absolute value greater than one-half and Re(s) between successive poles 4w, 4(w + 1) of f(s), with w an integer. The corresponding Laplace density is related to confluent hypergeometric functions. That density is shown to be positive for nonzero w other than -1. Those results are obtained without relying on any unproven hypothesis. They are used together with the Riemann hypothesis and hypotheses advanced by the author to obtain conditional results concerning the zeta-function. Those results are presented in Part I. Their proofs are derived in Parts III-V. A metric geometry expression of the positivity of the Laplace densities arising is established in Part VI.