On the irrationality of Ramanujan's mock theta functions and other q-series at an infinite number of points

TL;DR

This paper proves that various famous q-series and mock theta functions take on irrational values at infinitely many specific points, revealing new insights into their arithmetic nature.

Contribution

It demonstrates the irrationality of several key mock theta functions and q-series at infinitely many points, a novel result in the study of these special functions.

Findings

All considered mock theta functions are irrational at infinitely many points.

The results apply to functions of orders 3 and 5, including Rogers-Ramanujan series.

Irrationality holds at points q = ±1/n for n ≥ 2.

Abstract

We show that all of Ramanujan's mock theta functions of order 3, Watson's three additional mock theta functions of order 3, the Rogers-Ramanujan q-series, and 6 mock theta functions of order 5 take on irrational values at the points q=\pm 1/2,\pm 1/3,\pm 1/4,...