QEta 3.0

Abstract

QEta is a software package that implements an algorithm of Cristian-Silviu Radu to find Ramanujan-Kolberg identities. QEta includes the extension of Radu’s algorithm to the case of generalized eta-functions as given by Chen, Du, and Zhao in 2019. Furthermore, QEta can find a basis of relations of (generalized) Dedekind eta-functions as well as relations among dissections of eta-quotients.

Furthermore QEta provides a number of functions connected to the theory of modular functions.

QEta is programmed in the computer algebra system FriCAS.

Overview

The QEta package started at the end of 2015 with an implementation of the AB algorithm from the article “An algorithmic approach to Ramanujan-Kolberg identities” by Silviu Radu (2015) and the Samba algorithm from the article “Dancing Samba with Ramanujan Partition Congruences” by Ralf Hemmecke (2018).

In addition it implements the algorithm from the article “Construction of all Polynomial Relations among Dedekind Eta Functions of Level N” by Hemmecke and Radu to compute all polynomial relations of Dedekind eta-functions of a certain level (related website: Ideal of Relations of eta Functions).

The computations for the article “Construction of Modular Function Bases for Γ₀(121) related to p(11n+6)” by Hemmecke, Paule, and Radu were done with the QEta package. See also the related website.

In 2021 the extensions to Radu’s algorithm for generalized eta-quotients described in the the article “Finding Modular Functions for Ramanujan-Type Identities” by Chen, Du, and Zhao have also been included in QEta.

The underlying theory of the programs is described in the above articles which are also available as RISC reports 15-14, 16-06, 18-03. 19-10.

Among other result 16-06 contains a witness relation in terms of Dedekind eta functions for the Ramanujan congruence

\[p(11 n + 6) \equiv 0 \pmod{11}\]

for the partition function \(p(n)\) that is given through its generating series.

\[\sum_{n=0}^\infty p(n) q^n = \prod_{n=1}^\infty (1 − q^n)^{-1}\]

More deails and a description of how to work with QEta can be found in the QEtaTutorial.

The QEta package requires FriCAS 1.3.7. See QEta installation for installation instructions.