Special values of zeta functions and areas of triangles

Abstract

In this snapshot we give a glimpse of the interplay of special values of zeta functions and volumes of triangles. Special values of zeta functions and their generalizations arise in the computation of volumes of moduli spaces (for example of Abelian varieties) and their universal spaces. As a first example, we compute the special value of the Riemann zeta function (s) = 1 P n=1 n s at s = 2 and give its interpretation as the volume of the moduli space of elliptic curves. As a second example, we calculate a special value of the Mordell‐Tornheim zeta function using the Stern‐ Brocot tree. This example allows a geometric interpretation related to current research. 1 Some recollections of the rational numbers