2024/06/24 by Chen, Dandan, Chen, Rong, Garvan, Frank · 4 citations
#05A17 #11F33 #11F37 #11P83 #33D15 #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.2407.07107
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 ω(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.