An approach to the Lindel\"of Hypothesis for Dirichlet $L$-functions

TL;DR

The paper proposes a novel approach to the Lindel"of Hypothesis for Dirichlet L-functions using incomplete gamma functions and Touchard polynomials, offering a new perspective on their behavior at the critical line.

Contribution

It introduces a new representation of Dirichlet L-functions via incomplete gamma functions and Touchard polynomials, providing a fresh framework to analyze their properties.

Findings

Expressed Taylor coefficients in terms of Touchard polynomials

Reformulated the functional equation for Dirichlet L-functions

Provided a 'formula proof' of the Lindel"of hypothesis (not fully rigorous)

Abstract

The suggested approach is based on a known representation of Dirichlet $L$-functions via the incomplete gamma functions. Some properties of the Taylor coefficients of the lower incomplete gamma function at infinity seem to be new. Specifically, these coefficients can be expressed in terms of Touchard polynomials. Furthermore, these same coefficients can be used to reformulate the functional equation for Dirichlet $L$-functions. This relationship "explains"' why $\vert L_\chi(1/2+i t)\vert $ should be small. To present the new ideas in a nutshell, we start by giving (in Section 1) a "formula proof" of the Lindel\"of hypothesis. This is not a genuine proof, as we are not concerned with the convergence of our series nor do we justify changing the order of summation. In Section 2, we suggest some hypothetical ways of transforming the "proof" from Section 1 into a rigorous mathematical proof. Sections 3-5 contain some technical details and bibliographical references.