Joint extreme values of $L$-functions on and off the critical line

TL;DR

This paper demonstrates that multiple primitive GL(1) and GL(2) L-functions can simultaneously reach large values on the critical line without assuming the Riemann Hypothesis, and studies their joint distribution off the line.

Contribution

It provides unconditional results on simultaneous large values of primitive GL(1) and GL(2) L-functions and advances understanding of their joint distribution off the critical line.

Findings

Multiple primitive GL(1) and GL(2) L-functions can attain large values simultaneously on the critical line.

The joint distribution of GL(m) L-functions off the critical line is analyzed under zero-density estimates.

The paper improves upon previous results by removing the Riemann Hypothesis assumption for certain cases.

Abstract

It is shown that any number of distinct primitive $\mathrm{GL}(1)$ and $\mathrm{GL}(2)$ $L$-functions can simultaneously attain large values on the critical line. This is an unconditional improvement of a general result due to Heap and Li who have assumed the Riemann Hypothesis for more than three such $L$-functions. The joint distribution of $\mathrm{GL}(m)$ $L$-functions to the right of the critical line is also studied under certain zero-density estimates. In particular, we can partially recover results of Inoue and Li on Dirichlet $L$-functions and generally improve upon the work of Mahatab, Pa\'nkowski and Vatwani on the class of $L$-functions introduced by Selberg. The main machinery in both cases, on and off the critical line, is the resonance method of Soundararajan and Hilberdink/Voronin, respectively. On the critical line we additionally introduce a variation of Heath-Brown's method for the fractional moments of the Riemann zeta-function which makes it possible to avoid using any information on the zero distribution of $L$-functions whose degree is less than three.