Determinants of Laplacians, the Ray–Singer torsion on lens spaces and the Riemann zeta function

Abstract

Explicit expressions are obtained for the determinants of the Laplacians on zero- and one-forms for an infinite class of three dimensional lens spaces L(p,q). These expressions can be combined to obtain the Ray–Singer torsion of these lens spaces. As a consequence an infinite class of formulas is obtained for the Riemann zeta function ζ(3). The value of these determinants (and the torsion) grows as the size of the fundamental group of the lens space increases and this is also computed. The triviality of the torsion for just the three lens spaces L(6,1), L(10,3), and L(12,5) is also noted.