Exponential Sums with Additive Coefficients and its Consequences to Weighted Partitions

TL;DR

This paper studies the weighted partition function with strongly additive weights, deriving asymptotic formulas and analyzing exponential sums, thereby extending previous work to broader classes of functions including multiplicative functions.

Contribution

It establishes a new asymptotic formula for the weighted partition function with strongly additive weights, extending results to multiplicative functions and applying classical exponential sum estimates.

Findings

Derived an asymptotic formula for the weighted partition function

Extended analysis to multiplicative weight functions

Applied Montgomery-Vaughan estimate to exponential sums

Abstract

In this article, we consider the weighted partition function $p_f(n)$ given by the generating series $\sum_{n=1}^{\infty} p_f(n)z^n = \prod_{n\in\mathbb{N}^{*}}(1-z^n)^{-f(n)}$, 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.