The vortex-like behavior of the Riemann zeta function to the right of the critical strip

TL;DR

This paper investigates the vortex-like behavior of functions equivalent to the Riemann zeta function in the half-plane where real part exceeds 1, revealing complex argument variations and dense sets of points with specific argument properties.

Contribution

It introduces a new equivalence relation on exponential sums and demonstrates vortex-like argument behavior of equivalent functions near the line Re s=1.

Findings

Functions exhibit indefinite argument variation near Re s=1

Existence of dense sets of t where zeta curves make finite turns

Maximum abscissa for fixed argument is determined

Abstract

Based on an equivalence relation that was established recently on exponential sums, in this paper we study the class of functions that are equivalent to the Riemann zeta function in the half-strip $\{s\in\mathbb{C}:\operatorname{Re}s>1\}$. In connection with this class of functions, we first determine the value of the maximum abscissa from which the images of any function in it cannot take a prefixed argument. The main result shows that each of these functions experiments a vortex-like behavior in the sense that the main argument of its images varies indefinitely near the vertical line $\operatorname{Re}s=1$. In particular, regarding the Riemann zeta function $\zeta(s)$, for every $\sigma_0>1$ we can assure the existence of a relatively dense set of real numbers $\{t_m\}_{m\geq 1}$ such that the parametrized curve traced by the points $(\operatorname{Re}(\zeta(\sigma+it_m)),\operatorname{Im}(\zeta(\sigma+it_m)))$, with $\sigma\in(1,\sigma_0)$, makes a prefixed finite number of turns around the origin.