Scarcity of partition congruences on semiprime progressions
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866912637356343296 |
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| author | Ahlgren, Scott Beckwith, Olivia |
| author_facet | Ahlgren, Scott Beckwith, Olivia |
| contents | In recent work with Raum the authors considered congruences for the ordinary partition function $p(n)$ of the form $p(\ell Q^r n+β)\equiv 0\pmod\ell$ where $\ell, Q\geq 5$ are prime and $r\in \{1,2\}$, and proved a number of results which show that such congruences are scarce in a precise sense. Here we improve one of our results when $r=1$; in particular we prove (outside of trivial cases) that the set of primes $Q$ such that there exists $β\in \mathbb{Z}$ with $p(\ell Q n+β)\equiv 0\pmod \ell$ for all $n$ has density zero. The proof involves a modification of part of our previous argument and an application of a recent theorem of Dicks regarding modular forms of half-integral weight and level one modulo $\ell$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_19512 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Scarcity of partition congruences on semiprime progressions Ahlgren, Scott Beckwith, Olivia Number Theory In recent work with Raum the authors considered congruences for the ordinary partition function $p(n)$ of the form $p(\ell Q^r n+β)\equiv 0\pmod\ell$ where $\ell, Q\geq 5$ are prime and $r\in \{1,2\}$, and proved a number of results which show that such congruences are scarce in a precise sense. Here we improve one of our results when $r=1$; in particular we prove (outside of trivial cases) that the set of primes $Q$ such that there exists $β\in \mathbb{Z}$ with $p(\ell Q n+β)\equiv 0\pmod \ell$ for all $n$ has density zero. The proof involves a modification of part of our previous argument and an application of a recent theorem of Dicks regarding modular forms of half-integral weight and level one modulo $\ell$. |
| title | Scarcity of partition congruences on semiprime progressions |
| topic | Number Theory |
| url | https://arxiv.org/abs/2508.19512 |