Exponential Sums with Additive Coefficients and its Consequences to Weighted Partitions

Abstract

In this article, we consider the weighted partition function $p_f(n)$ given by the generating series ${\sum }_{n=1}^{\infty }{p}_f(n)z^n={\prod }_{n\in {\mathbb{N}}^{\ast }}(1-z^n{)}^{-f(n)}$, where we restrict the class of weight functions to strongly additive functions. Originally proposed in a paper by Yang, this problem was further examined by Debruyne and Tenenbaum for weight functions taking positive integer values. We establish an asymptotic formula for this generating series in a broader context, which notably can be used for the class of multiplicative functions. Moreover, we employ a classical result by Montgomery-Vaughan to estimate exponential sums with additive coefficients, supported on minor arcs.