Ramanujan congruences for overpartitions with restricted odd differences

TL;DR

This paper explores Ramanujan-type congruences for a specialized overpartition counting function, establishing the uniqueness of such congruences and using modular forms to generalize bounds on primes involved.

Contribution

It proves that only one Ramanujan congruence exists for overpartitions with restricted odd differences and extends bounds on primes for eta-quotient congruences.

Findings

Only one Ramanujan congruence exists for the function.

Two new congruences modulo 5 are provided.

A general theorem bounds primes for Ramanujan congruences in eta-quotients.

Abstract

We investigate Ramanujan congruences for the function which counts the overpartitions of n with restricted odd differences. In particular, we show that only one such congruence exists. Our method involves using the theory of modular forms to prove a more general theorem which bounds the number of primes possible for Ramanujan congruences in certain eta-quotients. This generalizes work done by Jonah Sinick. We also provide two congruences modulo 5 for this function.