Arithmetic properties of the partition function
Let \(p(n)\) be the number of partitions of the positive integer \(n\). For an odd prime \(l\), let \(\delta_l\) be such that \[ \delta_l = \frac{l^2-1}{24}. \] In this paper, the authors show that the only primes for which \[ p(ln-\delta_l)\equiv 0\pmod l \] are \(l=5, 7,\) and 11, corresponding to the famous Ramanujan's congruences for \(p(n)\), namely \[ \begin{aligned} p(5n-1) &\equiv 0\pmod 5, \\ p(7n-2)&\equiv 0\pmod 7, \quad\text{ and }\tag{\(*\)}\\ p(11n-5)&\equiv 0\pmod{11}.\end{aligned} \] More precisely, they prove that for \(l\geq 13\) \[ \sum_{n\geq 0} p(ln-\delta_l)q^n \not\equiv 0\pmod{l}. \] The proof of their result is very clever and unexpected (at least to the reviewer). The result follows from the properties of the filtration of a modular form modulo \(l\) (see Lemmas 2.1, 3.2 and 3.3 of the paper) discovered by J.-P. Serre and H. P. F. Swinnerton-Dyer. At the end of their argument, the authors show that if \[ \sum_{n\geq 0} p(ln-\delta_l)q^n \equiv 0\pmod{l}, \] then \(l|770\), which implies their main result. Let \(l\geq 5\) and define \(1\leq \beta_{l,j}\leq l^j-1\) by \[ 24\beta_{l,j}\equiv 1\pmod{l^j}. \] Define \[ k_{l,j} =\begin{cases} \frac{l^{j-1}(l-1)}{2}-\frac{1}{2}\left(\frac{24\beta_{l,j}-1}{l^j}+1\right)\quad &\text{if \(j\) is odd}, \\ l^{j-1}(l-1)-\frac{1}{2}\left(\frac{24\beta_{l,j}-1}{l^j}+1\right) \quad&\text{if \(j\) is even.} \end{cases} \] The second main result in this paper states that \[ \sum_{n\geq 0} p(l^jn+\beta_{l,j})q^n =\prod_{n\geq 1}(1-q^n)^{(24\beta_{l,j}-1)/l^j}F_{l,j}(q)\pmod{l^j} \] where \(F_{l,j}(q)\in \mathbb Zq\) is a modular form of weight \(k_{l,j}\) for \(\text{SL}_2(\mathbb{Z}).\) When \(j=1\), this coincides with a claim of Ramanujan (See section 15 of the paper [Ramanujan's unpublished manuscript on the partition and tau functions with proofs and commentary, Sémin. Lothar. Comb. 42, B42c (1999; Zbl 0932.11002)] by \textit{B. C. Berndt} and \textit{K. Ono}) and it implies that \[ \prod_{n\geq 1}(1-q^{ln})\sum_{n\geq 0}p(ln-\delta_l)q^n \] is congruent to a modular form of weight \(l-1\) on \(\text{SL}_2(\mathbb Z)\). The special case for \(j=1\) of their second result first appeared in a paper by \textit{K. S. Chua} [Arch. Math. 81, 11--21 (2003; Zbl 1047.11099)]. Note that when this result is applied to \(l=5,7,\) and 11, it immediately implies Ramanujan's congruences (\(*\)) since the dimensions of the spaces of cusp forms for \(\text{SL}_2(\mathbb Z)\) in these cases are 0.
scientific article; zbMATH DE number 2001026