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

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.