Lattice sums and Ramanujan's modular equations

Abstract

A modular equation of order n is essentially some algebraic relation between theta functions of arguments q and q n respectively. In his notebooks Ramanujan gave many such relations involving Lambert series, and Berndt [1] painstakingly collected these results, proved or verified them, and if necessary made corrections. However, these relations as given by Ramanujan appear in a haphazard way, and there seems to be no systematic way of ordering them or for that matter knowing whether the formulae are independent or complete. Here an attempt is made to arrange these results in a systematic fashion. It will also be demonstrated how new modular relations may be derived from those previously established. Indeed it will be shown how, using sign and Poisson transformations to be described in Chapter 6, each modular equation is essentially a set of either four or eight relations which can be generated from any one of the set by a group of simple transformations. Further it will also be shown how the use of character notation provides a shorthand for the lengthy Lambert series involved. A connection between binary quadratic forms and modular equations will be demonstrated. This will allow very simple proofs of certain modular and mixed modular equations to be executed. Finally it will be exhibited how the Mellin transforms of modular equations, in which q is replaced by e − t , lead to the evaluation of lattice sums.