St . Patrick ’ s College Dublin Institute for Advanced Studies Maynooth and 10 Burlington Road Ireland Dublin 4 Ireland

Starting with topological field theories we investigate the Ray-Singer analytic torsion in three dimensions. For the lens Spaces L(p; q) an explicit analytic continuation of the appropriate zeta functions is contructed and implemented. Among the results obtained are closed formulae for the individual determinants involved, the large p behaviour of the determinants and the torsion, as well as an infinite set of distinct formulae for ζ(3): the ordinary Riemann zeta function evaluated at s = 3. The torsion turns out to be trivial for and is, in general, greater than unity for large p and less than unity for a finite number of p and q. § 1. Introduction The torsion studied in this paper has its origins in the 1930's, cf. Franz [1], where it was combinatorially defined and used to distinguish various lens spaces from one another. Given a manifold M and a representation of its fundamental group π 1 (M) in a flat bundle E, this Reidemeister-Franz torsion is a real number which is defined as a particular product of ratio's of volume elements V i constructed from the cohomology groups H i (M ; E). Since volume elements are essentially determinants then, for any alternative definition of a determinant, an alternative definition of the torsion can be given. Now if one uses de Rham cohomology to compute H i (M ; E) then these determinants become determinants of Laplacians ∆ E p on p-forms with coefficients in E. But zeta functions for elliptic operators can be used to give finite values to such infinite dimensional determinants and so an analytic definition of the torsion results and this is the analytic torsion of Ray and Singer [2,3,4] given in the 1970's; furthermore this torsion was proved by them to be independent of the Riemannian metric used to define the Laplacian's ∆ E p. This analytic torsion coincided, for the case of lens spaces, with the combinatorially defined Reidemeister-Franz torsion. Finally Cheeger and Müller [5,6] independently proved that the analytic Ray-Singer torsion coincides with the combinatorial Reidemeister-Franz torsion in all cases. Infinite dimensional determinants also occur naturally in quantum field theories when computing correlation functions and partition functions. In 1978 Schwarz [7] showed how 1