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arXiv · 2606.19008

Explicit constructions for Ramanujan-type congruences

Wei Wang
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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.

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Wei Wang. 2026-06-17. Explicit constructions for Ramanujan-type congruences. https://arxiv.org/abs/2606.19008

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