Congruences for Andrews’ spt-function modulo powers of $5$, $7$ and $13$
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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.References
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Additional Information
- F. G. Garvan
- Affiliation: Department of Mathematics, University of Florida, Gainesville, Florida 32611-8105
- Email: fgarvan@ufl.edu
- Received by editor(s): November 20, 2010
- Published electronically: April 11, 2012
- Additional Notes: The author was supported in part by NSA Grant H98230-09-1-0051.
- © Copyright 2012
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 364 (2012), 4847-4873
- MSC (2010): Primary 11P83, 11F33, 11F37; Secondary 11P82, 05A15, 05A17
- DOI: https://doi.org/10.1090/S0002-9947-2012-05513-9
- MathSciNet review: 2922612
Dedicated: Dedicated to my friend and mentor Michael D. Hirschhorn on the occasion of his 63rd birthday