Proof of a conjecture of Keith on congruences of the reciprocal of a false theta function

TL;DR

This paper confirms Keith's conjecture on congruences of the reciprocal of a false theta function and extends the results to a more general setting, advancing understanding of these special functions.

Contribution

The paper proves Keith's conjecture and establishes a generalized congruence result for the coefficients of the reciprocal false theta function.

Findings

Keith's conjecture on congruences modulo 4 and 8 is confirmed.

A generalized congruence result is proved.

The work advances the theory of false theta functions and their reciprocals.

Abstract

Recently, Keith investigated reciprocals of false theta functions and proved some interesting results such as congruences, asymptotic bounds, and combinatorial identities. At the end of his paper, Keith posed a conjecture on congruences modulo 4 and 8 for the coefficients of the reciprocal of a false theta function. In this paper, we not only confirm Keith's conjecture, but also prove a generalized result.