# 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

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

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.

## Links¶

Bug reports: ralf@hemmecke.org