The partition function modulo prime powers (Q2839391)

Let \(p(n)\) denote the number of partitions of \(n\) and define the generating function \(P_{\ell}(b;z)\) by NEWLINE\[NEWLINE P_{\ell}(b;z) := \sum_{n \geq 0} p\left(\frac{\ell^b n+1}{24}\right)q^{n/24}. NEWLINE\]NEWLINE Here \(\ell \geq 5\) is prime, \(b \geq 0\) is an integer, and \(q := e^{2\pi iz}\). \textit{A. Folsom, Z. Kent} and \textit{K. Ono} [Adv. Math. 229, No. 3, 1586--1609 (2012; Zbl 1247.11131)] have shown that if \(m \geq 1\) is an integer, then there exists an integer \(b_{\ell}(m)\) with NEWLINE\[NEWLINE b_{\ell}(m) \leq 2\left(\Bigg \lfloor\frac{\ell-1}{12}\Bigg \rfloor + 2 \right)m - 3 NEWLINE\]NEWLINE such that the \(\mathbb{Z}/\ell^m\mathbb{Z}\)-module NEWLINE\[NEWLINE \text{Span}_{\mathbb{Z}/\ell^m\mathbb{Z}}\{P_{\ell}(b;z) : b \geq b_{\ell}(m), b \;\text{odd} \} NEWLINE\]NEWLINE has finite rank \(r_{\ell}(m)\), and that the same is true with ``odd'' replaced by ``even''.NEWLINENEWLINEThe main result of the paper under review is that NEWLINE\[NEWLINE b_{\ell}(m) \leq 2d_{\ell}+2m-1, NEWLINE\]NEWLINE where \(d_{\ell}\) is ``an explicitly calculable constant \dots related to the nullity of an operator \(D(\ell)\).'' The authors note the simple bound NEWLINE\[NEWLINE D_{\ell} \leq \Bigg \lfloor\frac{\ell-1}{12}\Bigg \rfloor, NEWLINE\]NEWLINE and they have computed that in fact \(d_{\ell} = 0\) for all primes \(5 \leq \ell \leq 1300\). For these primes one then has NEWLINE\[NEWLINE b_{\ell}(m) \leq 2m-1. NEWLINE\]NEWLINE An example of the kind of congruence implied by these results is NEWLINE\[NEWLINE p(53n+42) \equiv 22p(53^3n+117861) + 25p(53^5n + 331071432) \pmod{53}, NEWLINE\]NEWLINE which follows from the fact that \(b_{53}(1) = 1\) and \(r_{53}(1) = 2\).