TITLE: Visibility interpretation of Riemann zeta zeros on the critical line?
QUESTION [7 upvotes]: This is a long shot, but ...
The fraction of $\mathbb{Z}^2$ lattice points
visible from the origin
$1/\zeta(2)=6/\pi^2 \approx 61$%.
The fraction of $\mathbb{Z}^3$ lattice points visible
from the origin is $1/\zeta(3) \approx 83$%.
And this generalizes to arbitrary dimensions $1/\zeta(d)$.

Q. Is there some geometric/visibility interpretation of
  a zero of $\zeta(s)$ on the critical line, $\mbox{Re}(s)=1/2$?

I suspect not, but there are various notions of 
fractional dimension and
complex dimension
that might allow $\zeta(s)$ to have a type of geometric interpretation
in dimension $s$...?

REPLY [3 votes]: The zeros of the Riemann zeta function are telling you something about the error term in the counting problem.
Introduce the following zeta function
$$Z(s) = \sum_{ (a,b)=1 } 1/\max\{a,b\}^s.$$
Then an application of Mobius inverse shows that, modulo some constants, this is equal to
$$\zeta(s-1)/\zeta(s).$$
Perons formula can now be used to give an asymptotic formula for the corresponding counting problem. There is a pole of order $1$ at $s=2$ with residue $1/\zeta(2)$, which gives rise to the main term. There are also poles occuring at the zeros of the Riemann zeta function, which give rise to lower order terms in the corresponding asymptotic formula for the counting problem.
This is very similar to the fact that the zeros of the Riemann zeta function control the error term in the prime number theorem.