Explicit constructions for Ramanujan-type congruences

Abstract

For an integer-valued sequence $\{a(n)\}_{n\geq 0}$ and a prime $\ell$, a Ramanujan-type congruence is a relation of the form $a(\ell n-δ_{\ell})\equiv 0\pmod\ell$, where $δ_{\ell}$ is a specific shift. In this paper, we present explicit constructions of modular forms to establish Ramanujan-type congruences for a broad class of generating functions, including eta-quotients, weakly holomorphic modular forms, and mock modular forms. As applications, our explicit approach provides a unified framework to not only recover known congruences but also establish new non-congruence results and explicit congruences for various combinatorial and arithmetic functions.