Effective Congruences for Mock Theta Functions

TL;DR

This paper proves the existence of infinitely many linear congruences for Ramanujan's mock theta functions and provides an effective method to compute the smallest prime involved.

Contribution

It establishes new infinite congruences for mock theta coefficients and offers an effective way to determine the minimal prime for these congruences.

Findings

Infinitely many linear congruences for mock theta coefficients are proven.

An effective upper bound for the smallest prime in these congruences is provided.

The results build on and extend prior work by Lichtenstein and Treneer.

Abstract

Let M(q)=\sum c(n) q^n be one of Ramanujan's mock theta functions. We establish the existence of infinitely many linear congruences of the form c(An+B) \equiv 0 (mod \ell^j), where A is a multiple of \ell and an auxiliary prime p. Moreover, we give an effectively computable upper bound on the smallest such p for which these congruences hold. The effective nature of our results is based on prior works of Lichtenstein and Treneer.