Linear Incongruences for Generalized Eta-Quotients

TL;DR

This paper investigates linear incongruences in generalized eta-quotients, demonstrating that certain linear progressions do not produce linear congruences modulo primes, extending known results to broader classes of modular forms.

Contribution

It introduces a new method to analyze linear incongruences for generalized eta-quotients, generalizing classical results and applying to functions like Rogers-Ramanujan.

Findings

Linear progressions satisfying specific quadratic conditions do not produce linear congruences.

Extends classical eta-quotient results to generalized eta-quotients and weakly holomorphic modular forms.

Provides new insights into the structure of congruences for modular forms.

Abstract

For a given generalized eta-quotient, we show that linear progressions whose residues fulfill certain quadratic equations do not give rise to a linear congruence modulo any prime. This recovers known results for classical eta-quotients, especially the partition function, but also yields linear incongruences for more general weakly holomorphic modular forms like the Rogers-Ramanujan functions.