Computer-Assisted Proofs of Congruences for Multipartitions and Divisor Function Convolutions, based on Methods of Differential Algebra

TL;DR

This paper develops algebraic methods using Differential Algebra to prove new and existing congruences for multipartition and divisor functions, providing purely algebraic proofs for Ramanujan-type congruences and discovering new divisor function congruences.

Contribution

It introduces algebraic proofs of multipartition and divisor function congruences using Differential Algebra, including new divisor function results and algebraic proofs of known Ramanujan-type congruences.

Findings

Proved all Ramanujan-type congruences for q in {5,7,11} and a new one for q=17.

Discovered several new divisor function congruences.

Provided algebraic proofs for multipartition congruences previously verified only numerically.

Abstract

This paper provides algebraic proofs for several types of congruences involving the multipartition function and self-convolutions of the divisor function. Our computations use methods of Differential Algebra in $\mathbb{Z}/q\mathbb{Z}$, implemented in a couple of MAPLE programs available as ancillary files on arXiv. The first results of the paper are Ramanujan-type congruences of the form $p^{*k}(qn+r) \equiv_q 0$ and $\sigma^{*k}(qn+r) \equiv_q 0$, where $p(n)$ and $\sigma(n)$ are the partition and divisor functions, $q > 3$ is prime, and $^{*k}$ denotes $k^{th}$-order self-convolution. We prove all the valid congruences of this form for $q \in \{5, 7, 11\}$, including the three Ramanujan congruences, and a nontrivial one for $q = 17$. All such multipartition congruences have already been settled in principle up to a numerical verification due to D. Eichhorn and K. Ono via modular forms, but our proofs are purely algebraic. On the other hand, the majority of the divisor function congruences are new results. We then proceed to search for more general congruences modulo small primes, concerning linear combinations of $\sigma^{*k}(pn+r)$ for different values of $k$, as well as weighted convolutions of $p(n)$ and $\sigma(n)$ with polynomial weights. The paper ends with a few corollaries and extensions for the divisor function congruences, including proofs for three conjectures of N. C. Bonciocat.