The two dimensional distribution of values of $\zeta(1+it)$

TL;DR

This paper investigates the joint distribution of the argument and magnitude of the Riemann zeta function at 1+it, providing new results on its distribution in the complex plane, with analogous findings for L-functions.

Contribution

It establishes several results on the two-dimensional distribution function of (1+it), including joint distribution of argument and magnitude, and extends similar results to L(1, ) for non-principal characters.

Findings

Results on the distribution function of (1+it) in the complex plane.

Joint distribution results for (1+it) argument and magnitude.

Analogous distribution results for L(1,) over non-principal characters.

Abstract

We prove several results on the distribution function of $\zeta(1+it)$ in the complex plane, that is the joint distribution function of $\arg\zeta(1+it)$ and $|\zeta(1+it)|$. Similar results are also given for $L(1,\chi)$ (as $\chi$ varies over non-principal characters modulo a large prime $q$).