On the Riemann zeta-function, Part III

TL;DR

This paper constructs an odd meromorphic function from the Riemann zeta-function, deriving its partial fraction expansion under specific hypotheses, and uses it to obtain a Laplace transform representation within a certain vertical strip.

Contribution

It introduces a new function derived from the zeta-function and derives its partial fraction expansion using the Riemann hypothesis and additional hypotheses, advancing understanding of its properties.

Findings

Partial fraction expansion of the constructed function

Laplace transform representation within a vertical strip

Use of combined hypotheses to analyze zeta-related functions

Abstract

An odd meromorphic function f(s) is constructed from the Riemann zeta-function evaluated at one-half plus s. The partial fraction expansion, p(s), of f(s) is obtained using the conjunction of the Riemann hypothesis and hypotheses advanced by the author. That compound hypothesis and the expansion p(s) are employed in Part IV to derive the two-sided Laplace transform representation of f(s) on the open vertical strip of all s with real part between zero and four.