Twists and resonance of L-functions, I

TL;DR

This paper establishes fundamental analytic properties of nonlinear twists of L-functions in the extended Selberg class, solving the resonance problem for these functions across all degrees.

Contribution

It provides the first comprehensive analysis of meromorphic continuation, polar structure, and growth bounds for nonlinear twists of L-functions of any degree in the extended Selberg class.

Findings

Proves meromorphic continuation for all nonlinear twists with exponents ≤ 1/d.

Determines polar structures of these twists.

Establishes bounds on the growth of these twisted L-functions.

Abstract

We obtain the basic analytic properties, i.e. meromorphic continuation, polar structure and bounds for the order of growth, of all the nonlinear twists with exponents $\leq 1/d$ of the L-functions of any degree $d \geq 1$ in the extended Selberg class. In particular, this solves the resonance problem in all such cases.