Screw functions of Dirichlet series in the extended Selberg class

TL;DR

This paper introduces screw functions for Dirichlet series within the extended Selberg class and establishes their equivalence to the Grand Riemann Hypothesis through nonpositivity conditions.

Contribution

It defines screw functions for the extended Selberg class and links their properties directly to the Grand Riemann Hypothesis, providing a new analytical approach.

Findings

Screw functions are introduced for the extended Selberg class.

The Grand Riemann Hypothesis is equivalent to screw function nonpositivity.

A new criterion for the Riemann Hypothesis is established.

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.