  It oftens occurs that Taylor coefficients of (dimensionally regularized)
Feynman amplitudes $I$ with rational parameters, expanded at an integral
dimension $D= D_0$, are not only periods (Belkale, Brosnan, Bogner, Weinzierl)
but actually multiple zeta values (Broadhurst, Kreimer).
  In order to determine, at least heuristically, whether this is the case in
concrete instances, the philosophy of motives - more specifically, the theory
of mixed Tate motives - suggests an arithmetic approach (Kontsevich): counting
points of algebraic varieties related to $I$ modulo sufficiently many primes
$p$ and checking that the number of points varies polynomially in $p$.
  On the other hand, Kapranov has introduced a new "zeta function", the role of
which is precisely to "interpolate" between zeta functions of reductions modulo
different primes $p$.
  In this survey, we outline this circle of ideas and some of their recent
developments.
