On the Riemann zeta-function, Parts IV-V

TL;DR

This paper explores the properties of a meromorphic function derived from the Riemann zeta-function, deriving its Laplace transform representation and analyzing the first critical zero, under the assumption of the Riemann hypothesis and additional hypotheses.

Contribution

It introduces a two-sided Laplace transform representation of a function related to the zeta-function and provides a new expression for the first critical zero, z(1), under certain hypotheses.

Findings

Laplace density of the function is positive on a specific strip.

Derived an explicit expression for the first critical zero z(1).

Established relations between zeros and derivatives of the zeta-function.

Abstract

In Part I an odd meromorphic function f(s) has been constructed from the Riemann zeta-function evaluated at one-half plus s. The conjunction of the Riemann hypothesis and hypotheses advanced by the author in Part I is assumed. In Part IV we derive the two-sided Laplace transform representation of f(s) on the open vertical strip V of all s with real part between zero and four. An additional hypothesis is used to prove that the Laplace density of f(s) on the strip V is positive. Let z(n) be the nth critical zero of the Riemann zeta-function of positive imaginary part in order of magnitude thereof. In Part V an expression is derived for z(1). A relation is obtained of the pair z(n) and the first derivative thereat of the zeta-function to the preceding such pairs.