Large values of L-functions from the Selberg class

TL;DR

This paper establishes lower bounds for L-functions in the Selberg class using two different methods, and also provides upper bounds and additional applications, advancing understanding of these functions' size and behavior.

Contribution

It introduces two new theorems with different assumptions and proofs for lower bounds of Selberg class L-functions, improving previous results.

Findings

Both methods yield similar lower bounds, which are weaker than those for the Riemann zeta function.

The paper also derives upper bounds for these L-functions.

An additional application of Chen's theorem is presented.

Abstract

In the present paper we prove lower bounds for L-functions from the Selberg class, by this means improving earlier results obtained by the second author together with J\"orn Steuding. We formulate two theorems which use slightly different technical assumptions, and give two totally different proofs. The first proof uses the "resonance method", which was introduced by Soundararajan, while the second proof uses methods from Diophantine approximation which resemble those used by Montgomery. Interestingly, both methods lead to roughly the same lower bounds, which fall short of those known for the Riemann zeta function and seem to be difficult to be improved. Additionally to these results, we also prove upper bounds for L-functions in the Selberg class and present a further application of a theorem of Chen which is used in the Diophantine approximation method mentioned above.