Value-Distribution of the Riemann Zeta-Function along its Julia Lines

TL;DR

This paper investigates the distribution of the Riemann zeta-function's values at specific points related to its functional equation, revealing notable average behaviors along Julia lines associated with its essential singularity.

Contribution

It introduces a novel analysis of the zeta-function's values at $a$-points on Julia lines, highlighting their average distribution and behavior.

Findings

Values of $ ext{ exteta}( ext{ extdelta}_a)$ exhibit remarkable average patterns.

$a$-points cluster around the critical line, forming Julia lines.

The distribution of zeta-values along these lines shows distinctive statistical properties.

Abstract

For an arbitrary complex number $a\neq 0$ we consider the distribution of values of the Riemann zeta-function $\zeta$ at the $a$-points of the function $\Delta$ which appears in the functional equation $\zeta(s)=\Delta(s)\zeta(1-s)$. These $a$-points $\delta_a$ are clustered around the critical line $1/2+i\mathbb{R}$ which happens to be a Julia line for the essential singularity of $\zeta$ at infinity. We observe a remarkable average behaviour for the sequence of values $\zeta(\delta_a)$.