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arXiv · 2512.10658

Recurrence and congruences for the smallest parts function

Wei Wang
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Abstract

Let $\spt(n)$ be the number of smallest parts in the partitions of $n$. In this paper, we give some generalized Euler-like recursive formulas for the $\spt$ function in terms of Hecke trace of values of special twisted quadratic Dirichlet series. As a corollary, we give a closed form expression of the power series $\sum_{n\geq 0}\spt(\ell n-\delta_{\ell})q^n\pmod{\ell}$, $\delta_{\ell}:=(\ell^2-1)/24$, by Hecke traces for weight $\ell+1 $ cusp forms on $\SL_2(\mathbb{Z})$. We further establish an incongruence result for the $\spt$ function.

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Wei Wang. 2025-12-11. Recurrence and congruences for the smallest parts function. https://arxiv.org/abs/2512.10658

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