On the vanishing of some mock theta functions at odd roots of unity

TL;DR

This paper investigates the vanishing behavior of certain mock theta functions at odd roots of unity, proving they vanish beyond a certain point and conjecturing they do not vanish at any odd roots of unity.

Contribution

It establishes a proof that mock theta functions vanish at sufficiently large odd roots of unity and proposes a conjecture about their non-vanishing at all odd roots.

Findings

Mock theta functions vanish at all sufficiently large odd roots of unity.

A constant C exists such that for all odd n > C, the functions vanish at primitive n-th roots.

Conjecture: mock theta functions do not vanish at any odd roots of unity.

Abstract

We consider the problem of whether or not certain mock theta functions vanish at the roots of unity with an odd order. We prove for any such function $f(q)$ that there exists a constant $C>0$ such that for any odd integer $n>C$ the function $f(q)$ does vanish at the primitive $n$-th roots of unity. This leads us to conjecture that $f(q)$ does not vanish at the primitive $n$-th roots of unity for any odd positive integer $n$.