Title:Zeta function for convex domains
Speaker:Nikita Kalinin（Guangdong Technion Israel Institute of Technology）
Time: 07/03（Thursday）, 14:30-15:30
Venue: Gewu Building , 315
Abstract :Each irreducible fraction $p/q>0$ corresponds to a primitive vector $(p,q)\in\mathbb Z^2$ with $p,q>0$. Such a vector $(p,q)$ can be uniquely written as the sum of two primitive vectors $(a,b)$ and $(c,d)$ that span a parallelogram of oriented area one.
We present new summation formulae over the set of such parallelograms. These formulae depend explicitly on $a,b,c,d$ and thus define a summation over primitive vectors $(p,q)=(a+c,b+d)$ indirectly. Equivalently, these sums may be interpreted as running over pairs of consecutive Farey fractions $c/d$ and $a/b$, $ad-bc=1$.
The input for our formulae is the graph of a strictly concave function $g$. The terms are the areas of certain triangles formed by tangents to the graph of $g$. For $g$ being a parabola we recover the famous Mordell-Tornheim series (also called the Witten series).
Raising the terms in the above summation formula to the power $s$ we obtain an analytic function $F_g(s)$ which may be thought as the zeta function corresponding to the convex domain bounded by the graph of $g$. We show that for a concave $g$ in $C^2[0,1]$ the residue of $F_g(s)$ at $s=2/3$ is proportional to the affine length of the graph of $g$.
