On the zeros of the Riemann Zeta function

TL;DR

This paper introduces new representations of the Riemann zeta function, explores an equivalent formulation of the Riemann hypothesis, and provides exact zero information using a novel approach.

Contribution

It offers new integral and series representations of the zeta function and a different method to analyze its zeros, including an equivalent form of the Riemann hypothesis.

Findings

New series and integral representations of the zeta function

An equivalent formulation of the Riemann hypothesis

Exact zero information derived from the new approach

Abstract

This paper is divided into two independent parts. The first part presents new integral and series representations of the Riemaan zeta function. An equivalent formulation of the Riemann hypothesis is given and few results on this formulation are briefly outlined. The second part exposes a totally different approach. Using the new series representation of the zeta function of the first part, exact information on its zeros is provided.