, Volume 20, Issue 2, pp 215-251
Date: 29 Sep 2009

An algorithmic approach to Ramanujan’s congruences

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Abstract

In this paper we present an algorithm that takes as input a generating function of the form \(\prod_{\delta|M}\prod_{n=1}^{\infty}(1-q^{\delta n})^{r_{\delta}}=\sum_{n=0}^{\infty}a(n)q^{n}\) and three positive integers m,t,p, and which returns true if \(a(mn+t)\equiv0\pmod{p},n\geq0\), or false otherwise. Our method builds on work by Rademacher (Trans. Am. Math. Soc. 51(3):609–636, 1942), Kolberg (Math. Scand. 5:77–92, 1957), Sturm (Lecture Notes in Mathematics, pp. 275–280, Springer, Berlin/Heidelberg, 1987), Eichhorn and Ono (Proceedings for a Conference in Honor of Heini Halberstam, pp. 309–321, 1996).