Recurrence and congruences for the smallest parts function
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866911593396174848 |
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| author | Wang, Wei |
| author_facet | Wang, Wei |
| contents | 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 $\sum_{n\geq 0}\spt(\ell n-δ_{\ell})q^n\pmod{\ell}$, $δ_{\ell}:=(\ell^2-1)/24$, by Hecke traces for weight $\ell+1 $ cusp forms on $\SL_2(\mathbb{Z})$. We further establish an incongruence result for the $\spt$ function. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_10658 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Recurrence and congruences for the smallest parts function Wang, Wei Number Theory Combinatorics 11F37, 11P82, 11P83, 05A17 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 $\sum_{n\geq 0}\spt(\ell n-δ_{\ell})q^n\pmod{\ell}$, $δ_{\ell}:=(\ell^2-1)/24$, by Hecke traces for weight $\ell+1 $ cusp forms on $\SL_2(\mathbb{Z})$. We further establish an incongruence result for the $\spt$ function. |
| title | Recurrence and congruences for the smallest parts function |
| topic | Number Theory Combinatorics 11F37, 11P82, 11P83, 05A17 |
| url | https://arxiv.org/abs/2512.10658 |