This is a long shot, but ...


[The fraction of $\mathbb{Z}^2$ lattice points
visible from the origin](https://mathoverflow.net/q/151706/6094)
$1/\zeta(2)=6/\pi^2 \approx 61$%.
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*](http://en.wikipedia.org/wiki/Fractal_dimension) and
[*complex dimension*](http://en.wikipedia.org/wiki/Complex_dimension)
that might allow $\zeta(s)$ to have a type of geometric interpretation
in dimension $s$...?



Visibility interpretation of Riemann zeta zeros on the critical line?
The sort of picture that I have in my head looks like this: Whatever enumerative interpretation you give $\zeta(\sigma)$ for $\sigma >1$ of some (possibly weighted) objects, then information about $\zeta(\sigma)$ for $\sigma$ in the critical strip tells you how those objects are distributed or spaced from each other!


In the question at hand, while you can interpret the zeta function at positive integers as counting visible points, the Riemann hypothesis is equivalent to the statement that the "rays of vision" are equidistributed in some sense. Of course, the slopes of these "rays" taken in successive square regions give you the Farey sequence and the equivalent statement to the Riemann hypothesis I was alluding to is precisely the one by Franel/Landau mentioned [here](http://en.wikipedia.org/wiki/Farey_sequence#Riemann_hypothesis).


