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arXiv · 2412.21129

Exponential Sums with Additive Coefficients and its Consequences to Weighted Partitions

Madhuparna Das
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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.

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Madhuparna Das. 2024-12-30. Exponential Sums with Additive Coefficients and its Consequences to Weighted Partitions. https://arxiv.org/abs/2412.21129

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