Parity of the partition function in quadratic progressions
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866914071787339776 |
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| author | Ono, Ken |
| author_facet | Ono, Ken |
| contents | The parity of the partition function $p(n)$ remains strikingly mysterious. Beyond a handful of fragmentary results, essentially nothing is known about the distribution of parity. We prove a uniform result on quadratic progressions. If $1<D\equiv 23\pmod{24}$ is square-free and only divisible by primes $\ell\equiv 1, 7\pmod 8$, then both parities occur infinitely often among $$ p\left(\frac{Dm^2+1}{24}\right), $$ with $(m,6)=1.$ The argument takes place on the modular curve $X_0(6)$ and shows that parity along these thin orbits is \emph{not constant}. The proof connects classical identities for the partition generating function, through the method of (twisted) Borcherds products, to the arithmetic geometry of {\it ordinary} CM fibers on the Deligne-Rapoport model of $X_0(6)$ in characteristic 2. This result is a special case of a general theorem for the coefficients of suitable vector-valued weight 1/2 harmonic Maass forms that satisfy a "Heegner packet'' condition. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_09553 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Parity of the partition function in quadratic progressions Ono, Ken Number Theory Combinatorics 05A17, 11P82, 11G20 The parity of the partition function $p(n)$ remains strikingly mysterious. Beyond a handful of fragmentary results, essentially nothing is known about the distribution of parity. We prove a uniform result on quadratic progressions. If $1<D\equiv 23\pmod{24}$ is square-free and only divisible by primes $\ell\equiv 1, 7\pmod 8$, then both parities occur infinitely often among $$ p\left(\frac{Dm^2+1}{24}\right), $$ with $(m,6)=1.$ The argument takes place on the modular curve $X_0(6)$ and shows that parity along these thin orbits is \emph{not constant}. The proof connects classical identities for the partition generating function, through the method of (twisted) Borcherds products, to the arithmetic geometry of {\it ordinary} CM fibers on the Deligne-Rapoport model of $X_0(6)$ in characteristic 2. This result is a special case of a general theorem for the coefficients of suitable vector-valued weight 1/2 harmonic Maass forms that satisfy a "Heegner packet'' condition. |
| title | Parity of the partition function in quadratic progressions |
| topic | Number Theory Combinatorics 05A17, 11P82, 11G20 |
| url | https://arxiv.org/abs/2509.09553 |