Congruences modulo powers of $5$ and $7$ for the crank and rank parity functions and related mock theta functions
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.
- Publication:
-
arXiv e-prints
- Pub Date:
- June 2024
- DOI:
- 10.48550/arXiv.2407.07107
- arXiv:
- arXiv:2407.07107
- Bibcode:
- 2024arXiv240707107C
- Keywords:
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- Mathematics - Number Theory;
- Mathematics - Combinatorics;
- 05A17;
- 11F33;
- 11F37;
- 11P83;
- 33D15
- E-Print:
- 44 pages