Exponential Sums with Additive Coefficients and its Consequences to Weighted Partitions

2024/12/30 by Das, Madhuparna · 1 citation
#11L15 #11N60 #11P84 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2412.21129

Abstract

In this article, we consider the weighted partition function p_f(n) given by the generating series ∑_n=1^∞ p_f(n)zⁿ = ∏_n∈ℕ^*(1-zⁿ)^-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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