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RaduRK: Ramanujan-Kolberg Program
AuthorsRaduRK is a Mathematica implementation of an algorithm developed by Cristian-Silviu Radu. The algorithm takes as input an arithmetic sequence a(n) generated from a large class of q-Pochhammer quotients, together with a given arithmetic progression mn+j, and the level of a given congruence subgroup. The algorithm produces expressions for the generating function of a(mn+j) in terms of Q-linear combinations of Dedekind eta quotients which are modular over the subgroup. Identities of this form include famous results by Ramanujan which demonstrate the divisibility properties of p(5n+4) and p(7n+5).
The algorithm relies on certain powerful finiteness conditions imposed by the study of modular functions, and illustrates the utility of the subject to computational number theory.
The package has been developed by Nicolas Allen Smoot.
This program is free software: you can redistribute it and/or
modify it under the terms of the GNU General Public License as
published by the Free Software Foundation, either version 3 of the
License, or (at your option) any later version.
This program is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
GNU General Public License for more details.
You should have received a copy of the GNU General Public License
along with this program. If not, see https://www.gnu.org/licenses.
For using the package, download the file RaduRK.m, which can be found at
Place this file into a directory where Mathematica will find it.
For a demonstration of how to use the package see
For more ambitious examples, see:
The package requires 4ti2 and math4ti2.m, see
Instructions for the proper installation for these packages and RaduRK can be found in the following paper:
For details concerning the design of the algorithm, consult the following:
- S. Radu, "An Algorithmic Approach to Ramanujan's Congruences," Ramanujan Journal, 20, pp. 215-251 (2009).
- S. Radu, "An Algorithmic Approach to Ramanujan-Kolberg Identities," Journal of Symbolic Computation, 68, pp. 225-253 (2015).
Please report any bugs or other suggestions to Nicolas Smoot.