Simultaneous large values and dependence of Dirichlet $L$-functions in the critical strip

TL;DR

This paper investigates the joint behavior of Dirichlet L-functions in the critical strip, revealing their dependence and conditions under which they can simultaneously attain large values infinitely often.

Contribution

It demonstrates the dependence among Dirichlet L-functions and establishes their ability to achieve large values simultaneously infinitely often.

Findings

Dirichlet L-functions are dependent in the critical strip.

They do not behave as independent random variables.

Large simultaneous values occur infinitely often.

Abstract

We consider the joint value distribution of Dirichlet $L$-functions in the critical strip $\frac{1}{2} < \sigma < 1$. We show that the values of distinct Dirichlet $L$-functions are dependent in the sense that they do not behave like independently distributed random variables and they prevent each other from obtaining large values. Nevertheless, we show that distinct Dirichlet $L$-functions can achieve large values simultaneously infinitely often.