ARNOLD SEMINAR AT ICTP, TRIESTE. Uniform distribution of the nondivisible vectors in the crystalline space.
Starts 22 Feb 2008 14:30
Ends 22 Feb 2008 20:00
Central European Time
ICTP
Main Building Seminar Room
Strada Costiera, 11
I - 34151 Trieste (Italy)
A vector of the space of integers $\Z^n$ $(n>1)$ is called divisible if it is the product of some other vector of this space by an integer greater than 1.
The uniform distribution of a set $M$ in the $n$-dimensional integer space means that for any domain the number of points of $M$ in this domain, dilated homothetically with homothety coefficient $N$ is asymptotically proportional to the volume of this domain, multiplied by $N^n$. The proportionality coefficient is called the density of $M$.
THEOREM. The set $M$ of the nondivisible vectors of $\Z^n$ (n>1) is distributed there uniformly with density $1/\zeta(n)$, where $\zeta$ is the Euler zeta function.
Example. The density of the set of nondivisible vectors in the plane lattice equals $1/\zeta(2)=6/\pi^2\approx 2/3$.