the analytic continuation which gives $\ln \Det d^*d_{1}$. Hence we find that $$\eqalign{\ln \Det d^*d_{1}
&= {1\over p{\pi}^2}\zeta(3)
+{4\over p}\ln\left[2\sin({\pi\over p})\right]
+{4\over p}\sum_{l=1}^{(p-3)\over 2}
{(2l)}^2\ln\left[2\sin({2\pi l\over p})\right]\cr
&\qquad+{2(p-2)\over \pi}\int_{0}^{\pi\over p}dz z\cot(z)
-{4\over\pi}\sum_{l=1}^{(p-3)\over 2}
(2l)\int_{0}^{2\pi l\over p}dz z\cot(z)\cr
}\no$$ We are therefore now in a position to put these together and obtain an expression for the torsion. The first observation is that since $\tau_{+}(p,0)$ and $\tau_{-}(p,0)$ are both zero we have $$\tau(p,0)=0\no$$ This vanishing of $\tau(p,0)$ is related to the metric independence of the torsion something which has been established quite generally by Ray and Singer \[3\].

The torsion $T(p)$ itself is given by the difference of and . Combining these two expressions we find, upon a little simplification, that $T(p)$ is determined by the equation $$\eqalign{&\ln T(p)
=-{2\over p}\left[{(p^3-1)\over2\pi^2}\zeta(3)
+2\ln\left[2\sin({\pi\over p})\right]
+2\sum_{l=1}^{(p-3)\over 2}
4l^2\ln\left[2\sin({2l\pi\over p})\right] \right.\cr
&\qquad+{p^2\over\pi^2}\int_{0}^{\pi\over p}dz\,
z(z+{(p-2)\pi\over p})\cot(z)
+{p^2\over\pi^2}\int_{0}^{(p-1)\pi\over p}dz\,
z(z-{(p-2)\pi\over p})\cot(z)\cr
&\qquad\left.+2\sum_{l=1}^{(p-3)\over 2}
{p^2\over\pi^2}\int_{0}^{2l\pi\over p}dz\, z(z-{4l\pi\over p})\cot(z)
\right]\cr
}\no$$ Now the expression for the torsion from Ray’s calculation gave the alternative expression i.e. $$\ln T(p) = -4\ln\left[2 \sin({\pi\over p})\right]\no$$ We have verified that these two expressions and agree numerically, yet it is not transparent by inspection that this should be so; also using C.41 of appendix C 