An approach to the Lindelöf Hypothesis for Dirichlet $L$-functions

TL;DR

The paper proposes representing Dirichlet $L$-functions via incomplete gamma functions and links the functional equation to expansions in Touchard polynomials and associated $U_k$-polynomials. It offers a formal 'formula proof' of the Lindelöf hypothesis for $L_3(s)$, together with a program to render the argument rigorous through controlled truncations and gamma-tail acceleration, and then extends the framework to arbitrary primitive Dirichlet characters. Central to the approach is a 'key discovery' that certain weighted sums of $U_{2k}$-polynomials vanish, which mirrors the functional equation and yields a formal vanishing of $\xi_3(\tfrac{1}{2}+it)$; although not rigorous, this structure suggests directions for rigorous proofs via convergence control and series acceleration. By generalizing the identities to the full family of characters using $E_m$ and $F_{d,m}$ polynomials, the work provides a unified algebraic-analytic pathway to encode FE-induced cancellations, with potential implications for approaching Lindelöf in a novel analytic framework.

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_χ(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 Lindelöf 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.