Exact Russell-Type Modular Equations

TL;DR

This paper analyzes the coefficients of Russell-Type modular equations for the lambda function, providing uniform results for all odd primes without numerical evaluation, using a novel method based on multiplicative functions on partitions.

Contribution

It introduces a new method to study coefficients of modular equations using multiplicative functions, applicable uniformly across all odd primes, without relying on numerical q-expansion evaluations.

Findings

Coefficients are characterized uniformly for all odd primes.

The method avoids numerical q-expansion evaluations.

Potential extension to other modular equations.

Abstract

This paper provides some statistics for the coefficients of Russell- Type modular equations for the modular function, {\lambda}({\tau}). The results hold uniformly for all odd primes. They do not rely on any numerical evaluations of coefficients of q expansions of {\lambda}. The method relies on an internal structure of the coefficients of {\lambda} expressed in terms of multiplicative functions defined on integer partitions. The method may be extended to other types of modular equations.