A dynamic approach for the zeros of the Riemann zeta function - collision and repulsion

TL;DR

This paper introduces a new approach to the Riemann hypothesis by analyzing how zeros of approximate sections of the zeta function collide or repel each other as N varies, proposing a re-arrangement to prevent collisions and potentially prove RH.

Contribution

It proposes a novel dynamic framework studying zero interactions of the zeta function's approximations, including a re-arrangement method to avoid zero collisions, which could imply RH.

Findings

Zeros off the critical line result from collisions of consecutive zeros.

A re-arrangement of the series may prevent zero collisions.

Avoiding zero collisions could imply the Riemann hypothesis.

Abstract

For $N \in \mathbb{N}$ consider the $N$-th section of the approximate functional equation $$ \zeta_N(s)= \sum_{n =1 }^N B_n(s),$$ where $$ B_n(s)= \frac{1}{2} \left [ n^{-s} + \chi(s) \cdot n^{s-1} \right ].$$ Our aim in this work is to introduce a new approach for the Riemann hypothesis by studying the way pairs of consecutive zeros of $\zeta_N(s)$ change with respect to $N$. For the initial stage, it is known that the non-trivial zeros of $\zeta_1(s)$ all lie on the critical line $Re(s)=\frac{1}{2}$. In the region $2N \leq Im(s) \leq 2 \pi (N+1)$ the function $\zeta_N(s)$ serves as an approximation of $\zeta(s)$ itself, and it was conjectured by Spira that in this region $\zeta_N(s)$ also admits zeros only on the critical line. We show that the appearance of zeros of a section off the critical line can be realized as the result of two consecutive zeros meeting and pushing each other off the critical line as $N$ changes, a process to which we refer to as a collision of zeros. Based on a study of the properties of $\zeta_N(s)$, we suggest a way of re-arranging the order of summation of the elements $B_n(s)$ in $\zeta_{N}(s)$ with $N=\left [ \frac{Im(s)}{2} \right ]$ that is expected to avoid collisions altogether, we refer to such a re-arrangement as a repelling re-arrangement. In particular, establishing that the suggested repelling re-arrangement indeed avoids collisions for any pair of zeros would imply RH.