Ramanujan's type of product of theta functions and its applications

TL;DR

This paper develops new theorems for evaluating products of theta functions, applies them to compute Ramanujan's class invariants, and explores their algebraic properties in number fields.

Contribution

It introduces novel explicit evaluation methods for theta function products and uncovers new values of Ramanujan's class invariants.

Findings

Derived new explicit formulas for theta function products

Computed novel values of Ramanujan's class invariant g_n

Proved that certain theta products are units in algebraic number fields

Abstract

In this paper, we initiate a generous amount of new-found general theorems for explicit evaluations of product of the theta functions $b_{m, n}$ using Kronecker's limit formula and other various novel explicit evaluations that were introduced thereupon. Also, we have obtained a few novel values of Ramanujan's class invariant $g_n$ using new-found values of $b_{m, n}$. Eventually, we have shown that $b_{m, n}$ is the unit in certain algebraic number fields.