That's not really the geometric interpretation of the Riemann zeta function, but rather the series \sum_{k=1}^\infty 1/k^n. Riemann zeta function makes sense not only in positive integers greater than 1, but also on the almost whole complex plane, and identifying complex zeros of zeta function is of main interest.
"If a lattice point is selected at random in two dimensions, the probability that it is visible from the origin is 6/π^2. This is also the probability that two integers picked at random are relatively prime. If a lattice point is picked at random in n dimensions, the probability that it is visible from the origin is 1/ζ(n), where ζ(n) is the Riemann zeta function."
[1] http://mathworld.wolfram.com/VisiblePoint.html