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Multiple standard twists of $L$-functions

Jerzy Kaczorowski, Alberto Perelli

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 $F(s,α)$ of an $L$-function $F$ are well-understood. For example, it has poles when the positive number $α$ 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 ${\mathbf F}({\mathbf s},α)$. 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 ${\mathbf F}({\mathbf s},α)$. There are also significant differences; for instance, in the structure of the singularities.

Multiple standard twists of $L$-functions

Abstract

The standard twist of -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 of an -function are well-understood. For example, it has poles when the positive number belongs to the so-called spectrum of , and is entire otherwise. In this paper, for a given set of -functions and , we consider the multiple standard twist . This is defined initially on a certain half-space of , 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 . There are also significant differences; for instance, in the structure of the singularities.
Paper Structure (4 sections, 10 theorems, 153 equations)

This paper contains 4 sections, 10 theorems, 153 equations.

Table of Contents

  1. Introduction
  2. Definitions and lemmas
  3. Proof of Theorem \ref{['T1']}
  4. Proof of Theorem \ref{['T2']}

Key Result

Theorem 1

For $N\geq2$ we have (i) if $\alpha\not\in {\rm Spec}({\boldsymbol{F}})$, then ${\boldsymbol{F}}( {\boldsymbol{s}}, \alpha)$ has analytic continuation to an entire function on $\mathbb C^N$; (ii) if $\alpha\in {\rm Spec}({\boldsymbol{F}})$, then it has analytic continuation to a holomorphic function and for every $\ell\geq 0$ and ${\boldsymbol{s}}\in H^*_\ell$ the limit exists and is holomorphic

Theorems & Definitions (18)

  • Theorem 1
  • Theorem 2
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • ...and 8 more