Exact Russell-Type 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.