A Dirichlet Generating Function for the Coefficients of Euler's Pentagonal Number Theorem
Friedjof Tellkamp
TL;DR
This work constructs a Bromwich-type Dirichlet generating function $D^*(s)$ for the coefficients $a_n$ in Euler's pentagonal number theorem, enabling analytic continuation of the classical $D(s)=\sum a_n n^{-s}$ to all complex $s$ and revealing its entire nature. The main tool is an explicit integral representation $D^*(s)=\frac{1}{2\pi i}\int F(z)\,u(z)^{-s}\,dz$, where $F(z)$ is even and $2\pi$-periodic and the residues at real poles generate the generalized pentagonal-number pattern of $a_n$. The paper also derives the asymptotic behavior $D^*(s)\sim 2\sqrt{3}\cdot 6^{-s}\zeta(2s)$ as $s\to -\infty$, provides Perron-type formulas for partial sums, and yields integral representations for the Euler function $\phi(q)$ and the Dedekind eta function $\eta(\tau)$ via Hankel contours. Finally, for positive integers $k$, an explicit finite-sum formula expresses $D(k)$ in terms of Bernoulli numbers and Glaisher's $G^*$-numbers, tying the DGF to classical arithmetic constants and enabling concrete evaluations, including numerical values for $D(1)$, $D(2)$, and $D(3)$. This framework clarifies the analytic structure of the pentagonal-number sequence and provides a toolset for further exploration of modular-form-related coefficients.
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 $φ(q)$ and the Dedekind eta function $η(τ)$. Moreover, we obtain an explicit formula for the Dirichlet generating function at each positive integer, expressed as a finite sum.