Arguments Related to the Riemann Hypothesis: New Methods and Results

TL;DR

This paper explores new equivalences and necessary conditions related to the Riemann hypothesis, using analytic, experimental, and graphical methods to deepen understanding of the zeros of the zeta function.

Contribution

It establishes the equivalence of certain propositions related to the zeros of the zeta function and introduces a new necessary condition for the Riemann hypothesis.

Findings

Proves the equivalence between the Riemann hypothesis and a condition on zeros of a derivative.

Shows that the Riemann hypothesis implies a new necessary condition for the distribution of zeros.

Uses a combination of analytic, experimental, and graphical techniques to derive results.

Abstract

Four propositions are considered concerning the relationship between the zeros of two combinations of the Riemann zeta function and the function itself. The first is the Riemann hypothesis, while the second relates to the zeros of a derivative function. It is proved that these are equivalent, and that, if the Riemann hypothesis holds, then all zeros of the zeta function on the critical line are simple. The Riemann hypothesis is then shown to imply the third proposition holds, this being a new necessary condition for the Riemann hypothesis. The third proposition is shown to be equivalent to the fourth, and either is shown to yield the result that the distribution of zeros on the critical line of $\zeta (s)$ is that given by the Riemann hypothesis. The results given are obtained from a combination of analytic arguments, experimental mathematical techniques and graphical reasoning.