Congruences for Ramanujan’s f and ω functions via generalized Borcherds products

Abstract

Bruinier and Ono recently developed the theory of generalized Borcherds products, which uses coefficients of certain Maass forms as exponents in infinite product expansions of meromorphic modular forms. Using this, one can use classical results on congruences of modular forms to obtain congruences for Maass forms. In this note we work out the example of Ramanujan's mock theta functions f and ω in detail.2010 Mathematics Subject Classification. 11F03, 11F33.

Parity of coefficients of mock theta functions

“…Several congruences for c(ω (3) ; n) and c(ν (3) ; n) modulo 11 were also presented in [46]. For more congruences satisfied by coefficients of mock theta functions, see the works of Berg et al [7], Brietzke, Silva and Sellers [12], Chern and Wang [16], Mao [37] and Xia [49], for example. For any formal power series g(q) as in (1.10), we define its type modulo a positive integer as follows.…”

Overpartitions related to the mock theta function $ω(q)$

“…Since Ramanujan introduced mock theta functions in his last letter to Hardy in 1920, they have been the subject of intense study for many decades. Along with his third order mock theta function f (q), there are many studies on the mock theta function ω(q) := ∞ n=1 q 2n 2 +2n (q; q 2 ) 2 n+1 in the literature [12], [16], [19], [27]. Throughout the paper, we adopt the following q-series notation:…”