Explicit congruences for the partition function modulo every prime
Let \(p(n)\) denote the partition function. Let the prime \(m\geq 13\). Let \(q= e^{2\pi iz}\), \(\delta_m\equiv 24^{-1}\pmod m\), \(0<\delta_m< m\), \(r_m\equiv -m\pmod{24}\), \(0< r_m< 24\). Let \(\eta(z)\) denote the Dedekind eta function. Let \(\lambda_m\) be the least integer such that \(24\lambda_m> m\). The author proves that \[ \sum^\infty_{n=0} p(nm+ \delta_m) q^{24n+ r_m}\equiv \eta^{r_m}(24z)\varphi_m(24z) \pmod m,\tag{1} \] where \(\phi_m(z)\) is an explicitly computable holomorphic function of weight \(m- 12\lambda_m- 1\) for \(\text{SL}_2(z)\). For each prime, \(m\), such that \(13\leq m\leq 97\), the author lists the quantities \(r_m\), \(\delta_m\), \(\lambda_m\), \(m- 12\lambda_m- 1\), \(\eta^{r_m}\phi_m\). He also conjectures an extension of (1) to powers of \(m\). This work extends prior results by Ahlgren, Ono, and Weaver.
scientific article; zbMATH DE number 2051572