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arXiv · 2407.07107

Congruences modulo powers of $5$ and $7$ for the crank and rank parity functions and related mock theta functions

Dandan Chen·Rong Chen·Frank Garvan
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Abstract

It is well known that Ramanujan conjectured congruences modulo powers of $5$, $7$ and and $11$ for the partition function. These were subsequently proved by Watson (1938) and Atkin (1967). In 2009 Choi, Kang, and Lovejoy proved congruences modulo powers of $5$ for the crank parity function. The generating function for the analogous rank parity function is $f(q)$, the first example of a mock theta function that Ramanujan mentioned in his last letter to Hardy. Recently we proved congruences modulo powers of $5$ for the rank parity function, and here we extend these congruences for powers of $7$. We also show how these congruences imply congruences modulo powers of $5$ and $7$ for the coefficients of the related third order mock theta function $\omega(q)$, using Atkin-Lehner involutions and transformation results of Zwegers. Finally we a prove a family of congruences modulo powers of $7$ for the crank parity function.

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Dandan Chen, Rong Chen, Frank Garvan. 2024-06-24. Congruences modulo powers of $5$ and $7$ for the crank and rank parity functions and related mock theta functions. https://arxiv.org/abs/2407.07107

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