A Dirichlet Generating Function for the Coefficients of Euler's Pentagonal Number Theorem

TL;DR

This paper introduces an integral representation for the Dirichlet generating function of Euler's pentagonal number theorem coefficients, enabling analytic continuation and asymptotic analysis, and provides explicit formulas for special cases.

Contribution

It establishes a novel integral representation and analytic continuation for the Dirichlet generating function related to Euler's pentagonal numbers, filling a gap in existing literature.

Findings

Derived an integral representation for the Dirichlet generating function.

Achieved analytic continuation to the entire complex plane.

Provided explicit formulas at positive integers.

Abstract

We establish an integral representation for the Dirichlet generating function of the coefficients of Euler's pentagonal number theorem. The Bromwich-type integral enables analytic continuation to the entire complex plane, filling a gap in the literature and providing a new framework for studying the sequence's analytic structure. Furthermore, we derive the asymptotic behavior as the variable tends to negative infinity, and give integral representations for the Euler function $\phi(q)$ and the Dedekind eta function $\eta(\tau)$. Moreover, we obtain an explicit formula for the Dirichlet generating function at each positive integer, expressed as a finite sum.