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Screw functions of Dirichlet series in the extended Selberg class

Masatoshi Suzuki

TL;DR

This paper generalizes the screw-function framework to the extended Selberg class by defining a screw function $g_F(t)$ for $F \in \mathcal{S}^{\sharp}$. It proves that the Grand Riemann Hypothesis for $F$ is equivalent to the nonpositivity of $\Re(-g_F(t))$ for large $t$, assuming no real zeros besides possibly at $s=\tfrac{1}{2}$. The authors derive zero-free formulas for $g_F$ in the semi-extended class $\mathcal{S}^{\sharp\flat}$ via Lerch transcendent representations, giving a representation of $-g_F(t)$ that does not rely on the zeros. Connections to explicit formulas, Weil distributions, and Li-type criteria are discussed, and the work outlines extensions to broader Selberg-class families and potential zero-free-region consequences.

Abstract

We introduce screw functions for Dirichlet series in the extended Selberg class. Then we prove that the Grand Riemann Hypothesis for a member of the extended Selberg class is equivalent to the nonpositivity of the corresponding screw function.

Screw functions of Dirichlet series in the extended Selberg class

TL;DR

This paper generalizes the screw-function framework to the extended Selberg class by defining a screw function for . It proves that the Grand Riemann Hypothesis for is equivalent to the nonpositivity of for large , assuming no real zeros besides possibly at . The authors derive zero-free formulas for in the semi-extended class via Lerch transcendent representations, giving a representation of that does not rely on the zeros. Connections to explicit formulas, Weil distributions, and Li-type criteria are discussed, and the work outlines extensions to broader Selberg-class families and potential zero-free-region consequences.

Abstract

We introduce screw functions for Dirichlet series in the extended Selberg class. Then we prove that the Grand Riemann Hypothesis for a member of the extended Selberg class is equivalent to the nonpositivity of the corresponding screw function.
Paper Structure (7 sections, 4 theorems, 28 equations)

This paper contains 7 sections, 4 theorems, 28 equations.

Table of Contents

  1. Introduction
  2. The Selberg class and explicit formulas
  3. Proof of the main results
  4. Proof of Theorem \ref{['thm_101']}
  5. Proof of Corollary \ref{['cor_1']}
  6. Zero-free formulas of screw functions
  7. Complements

Key Result

Theorem 1.1

Let $F$ be a member of $\mathcal{S}^{\sharp}$ and let $g_F(t)$ be the screw function defined by eq_101. We assume that $F(s)$ has no real zeros except for the possible zero at $s=1/2$. Then, the GRH for $F$ is true if and only if $\Re(-g_F(t)) \geq 0$ for all $t \geq t_0$ for some $t_0 \geq 0$.

Theorems & Definitions (5)

  • Theorem 1.1
  • Corollary 1.1
  • Theorem 4.1
  • proof
  • Theorem 5.1