Congruences for Andrews’ spt-function modulo powers of 5, 7 and 13

Garvan, F.

doi:10.1090/S0002-9947-2012-05513-9

Congruences for Andrews’ spt-function modulo powers of $5$, $7$ and $13$
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by F. G. Garvan PDF

Trans. Amer. Math. Soc. 364 (2012), 4847-4873 Request permission

Abstract:

Congruences are found modulo powers of $5$, $7$ and $13$ for Andrews’ smallest parts partition function $\mbox {spt}(n)$. These congruences are reminiscent of Ramanujan’s partition congruences modulo powers of $5$, $7$ and $11$. Recently, Ono proved explicit Ramanujan-type congruences for $\mbox {spt}(n)$ modulo $\ell$ for all primes $\ell \ge 5$ which were conjectured earlier by the author. We extend Ono’s method to handle the powers of $5$, $7$ and $13$ congruences. We need the theory of weak Maass forms as well as certain classical modular equations for the Dedekind eta-function.

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