On the Value Distribution of Two Dirichlet L-functions

TL;DR

This paper investigates the value distribution of two Dirichlet L-functions at the Riemann zeros, showing that their values are often linearly independent and have different arguments, especially off the critical line.

Contribution

It demonstrates the linear independence of the values of two Dirichlet L-functions at Riemann zeros off the critical line and quantifies their differences on the critical line.

Findings

A positive proportion of zeros have linearly independent L-values.

Values of the two L-functions differ in argument for many zeros.

On the critical line, a positive proportion of zeros have different L-values.

Abstract

We look at the values of two Dirichlet $L$-functions at the Riemann zeros (or a horizontal shift of them). Off the critical line we show that for a positive proportion of these points the pairs of values of the two $L$-functions are linearly independent over $\mathbb{R}$, which, in particular, means that their arguments are different. On the critical line we show that, up to height $T$, the values are different for $cT$ of the Riemann zeros for some positive $c$.