Multiple standard twists of $L$-functions

TL;DR

This paper extends the theory of standard twists of $L$-functions to multiple twists involving several functions, describing their meromorphic continuation and analyzing their singularities and spectra.

Contribution

It introduces the concept of multiple standard twists of $L$-functions, providing a detailed description of their meromorphic continuation and pole structure in higher dimensions.

Findings

Meromorphic continuation of multiple twists to the whole space

Spectral analysis of poles related to the set of functions

Structural differences in singularities compared to one-dimensional case

Abstract

The standard twist of $L$-functions plays a fundamental role in the Selberg class theory. It is defined as an absolutely convergent Dirichlet series and admits meromorphic continuation beyond the half-plane of absolute convergence. Nowadays, the analytic properties of the standard twist $F(s,\alpha)$ of an $L$-function $F$ are well-understood. For example, it has poles when the positive number $\alpha$ belongs to the so-called spectrum of $F$, and is entire otherwise. In this paper, for a given set ${\mathbf F}=\{F_1,\dots,F_N\}$ of $L$-functions and ${\mathbf s}\in{\mathbb C}^N$, we consider the multiple standard twist ${\mathbf F}({\mathbf s},\alpha)$. This is defined initially on a certain half-space of ${\mathbb C}^N$, and we describe its meromorphic continuation to the whole space. Results in the multidimensional case are, in many ways, analogous to those in the one-dimensional case. In particular, the spectrum of a multiple standard twist is relevant to the description of the set of poles of ${\mathbf F}({\mathbf s},\alpha)$. There are also significant differences; for instance, in the structure of the singularities.