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 $a(\ell n-\delta_{\ell})\equiv 0\pmod\ell$, where $\delta_{\ell}$ 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.
From: Wei Wang [view email]
[v1]
Wed, 17 Jun 2026 12:34:13 UTC (23 KB)
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