Ramanujan--Fine integrals for level 10

TL;DR

This paper explores eta quotients at level 10, establishing integral evaluations and classifying when such quotients are derivatives of integer-coefficient power series, with implications for mathematical analysis and number theory.

Contribution

It introduces a classification of eta quotients at level 10 and provides explicit integral evaluations, advancing understanding of their structure and properties.

Findings

Classified eta quotients as derivatives of integer-coefficient series at level 10

Established explicit integral evaluations involving eta quotients

Conjectured the completeness of the classification for level 10

Abstract

We investigate the question of when an eta quotient is a derivative of a formal power series with integer coefficients and present an analysis in the case of level 10. As a consequence, we establish and classify an infinite number of integral evaluations such as $$ \int_0^{e^{-2\pi/\sqrt{10}}} q\prod_{j=1}^\infty \frac{(1-q^j)^3(1-q^{10j})^8}{(1-q^{5j})^7} \text{d} q = \frac14\left(\sqrt{10-4\sqrt{5}}-1\right). $$ We describe how the results were found and give reasons for why it is reasonable to conjecture that the list is complete for level 10.