On the Riemann zeta-function, Part I: Outline

TL;DR

This paper explores the Riemann zeta-function under the assumption of RH and other hypotheses, deriving relations involving its zeros and their derivatives, and establishing positivity of a Laplace density that implies RH and zero simplicity.

Contribution

It introduces a novel relation connecting zeta zeros, derivatives, and specific values, and proves the positivity of a Laplace density that implies RH and zero simplicity.

Findings

Derived a relation linking zeta zeros and derivatives to specific function values.

Proved the Laplace density associated with zeta has positive values.

Established that positivity of the density implies RH and zero simplicity.

Abstract

Results of a multipart work are outlined. Use is made therein of the conjunction of the Riemann hypothesis, RH, and hypotheses advanced by the author. Let z(n) be the nth nonreal zero of the Riemann zeta-function with positive imaginary part in order of magnitude thereof. A relation is obtained, of the pair z(n) and the first derivative thereat of the zeta-function, to the preceding such pairs and the values of zeta at the points one-half plus a nonnegative multiple of four. That relation is derived from two forms of the density of the Laplace representation, on a certain vertical strip, of a meromorphic function constructed from zeta. Specific functions which play a central role therein are proven to have analytic extensions to the entire complex plane. It is established that the Laplace density is positive. That positivity implies RH and that each nonreal zero of zeta is simple. A metric geometry expression of the positivity of the density is derived. An analogous context is delineated relative to Dirichlet L-functions and the Ramanujan tau Dirichlet function.