A simple evaluation of a theta value, the Kronecker limit formula and a formula of Ramanujan

Abstract

We evaluate the classic sum n∈Z e−πn2 . The novelty of our approach is that it does not require any prior knowledge about modular forms, elliptic functions or analytic continuation. Even the function, in terms of which the result is expressed, only appears as a complex function in the computation of a real integral by the residue theorem. Another contribution of this note is to provide a very simple proof of the Kronecker limit formula. Finally, employing the evaluation of the sum and some other ideas, we also obtain an undemanding proof of one of the most emblematic formulas of Ramanujan