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**Determinants of Laplacians, the Ray-Singer Torsion on Lens Spaces**

**and the Riemann zeta function**

1.25

by

1.25

Charles Nash and D. J. O’ Connor

Department of Mathematical PhysicsSchool of Theoretical Physics

St. Patrick’s CollegeDublin Institute for Advanced Studies

Maynooth10 Burlington Road

IrelandDublin 4

Ireland

3[[**Abstract**]{}: We obtain explicit expressions 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 we obtain an infinite class of formulae for the Riemann zeta function $\zeta(3)$. 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.]{} Topological phenomena are now known to 