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Congruences modulo powers of $5$ and $7$ for the crank and rank parity functions and related mock theta functions
- Publication Year :
- 2024
-
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.<br />Comment: 44 pages
Details
- Database :
- arXiv
- Publication Type :
- Report
- Accession number :
- edsarx.2407.07107
- Document Type :
- Working Paper