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arXiv · 2610.13864

On Divisor Function Polynomials and Dirichlet Series

Martin N. Huxley·Matthew C. Lettington·Karl Michael Schmidt
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Abstract

We find the Dirichlet series and convergence bounds for the counting function of sum systems (collections of sets of non-negative integers whose Minkowski sum is an arithmetic progression). This motivates the definition and study of divisor function polynomials, which are generating functions for the non-trivial divisor function and for the balanced divisor function that counts, up to a sign, non-trivial ordered factorisations into squarefree factors. These divisor function polynomials turn out to be intimately connected between each other and with Bernoulli numbers and ordered Bell polynomials.

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Martin N. Huxley, Matthew C. Lettington, Karl Michael Schmidt. 2026-10-09. On Divisor Function Polynomials and Dirichlet Series. https://arxiv.org/abs/2610.13864

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