Determinants of Laplacians and a second limit formula in \(\text{GL}(3)\)
The aim of this paper is twofold. First, we consider the Eisenstein series on \(\text{GL}(3)\), twisted by an additive character, and evaluate it at \(s=3/2\). This is a rank two version of Kronecker's celebrated second limit formula, and thus involves functions which are \(\text{GL}(3)\) analogs of the absolute value of the eta and theta functions. We then proceed to use this formula in order to give a precise expression for the determinant of the Laplace operator on three dimensional flat tori. We find that this determinant, as a function on the moduli space of such tori, is a harmonic function there.
scientific article; zbMATH DE number 3981262