One can define a random walk on a hypercubic lattice in a space of int
eger dimension D. For such a process formulas can be derived that expr
ess the probability of certain events, such as the chance of returning
to the origin after a given number of time steps. These formulas are
physically meaningful for integer values of D. However, these formulas
are unacceptable as probabilities when continued to noninteger D beca
use they give values that can be greater than 1 or less than 0. In thi
s paper a different kind of random walk is proposed which gives accept
able probabilities for all real values of D. This D-dimensional random
walk is defined on a rotationally symmetric geometry consisting of co
ncentric spheres. The exact result is given for the probability of ret
urning to the origin for all values of D in terms of the Riemann zeta
function. This result has a number-theoretic interpretation.