  In this paper we show novel underlying connections between fractional powers
of the Laplacian on the unit sphere and functions from analytic number theory
and differential geometry, like the Hurwitz zeta function and the
Minakshisundaram zeta function. Inspired by Minakshisundaram's ideas, we find a
precise pointwise description of $(-\Delta_{\mathbb{S}^{n-1}})^s u(x)$ in terms
of fractional powers of the Dirichlet-to-Neumann map on the sphere. The Poisson
kernel for the unit ball will be essential for this part of the analysis. On
the other hand, by using the heat semigroup on the sphere, additional pointwise
integro-differential formulas are obtained. Finally, we prove a
characterization with a local extension problem and the interior Harnack
inequality.
