# An Asymptotic Form of the Generating Function $\prod_{k=1}^\infty (1+x^k/k)$

## TL;DR

This paper investigates the asymptotic behavior of the generating function product involving harmonic terms, showing the convergence of its coefficients to a positive limit expressed through the Riemann zeta function.

## Contribution

It provides an explicit expression for the limit of the sequence generated by an infinite product involving harmonic terms, linking it to the Riemann zeta function.

## Key findings

- The sequence of coefficients converges to a positive constant C.
- The constant C is expressed as an exponential involving the Riemann zeta function.
- The generating function's asymptotic form is characterized explicitly.

## Abstract

It is shown that the sequence of rational numbers $r(k)$ generated by the ordinary generating function $\prod_{k=1}^\infty (1+x^k/k)$ converges to a limit $C > 0$. $C$ can be expressed as $C = \exp\Bigl(-\sum_{k = 2}^\infty \frac{(-1)^k}{k}\ \zeta(k) \Bigr)$ where $\zeta()$ denotes the Riemann zeta function.