On a Two-Variable Zeta Function for Number Fields

This paper studies a two-variable zeta function ZK(w, s) attached to an algebraic number field K, introduced by van der Geer and Schoof [11], which is based on an analogue of the RiemannRoch theorem for number fields using Arakelov divisors. When w = 1 this function becomes the completed Dedekind zeta function ζ̂K(s) of the field K. The function is an meromorphic function of two complex variables with polar divisor s(w − s), and it satisfies the functional equation ZK(w, s) = ZK(w,w− s). We consider the special case K = Q, where for w = 1 this function is ζ̂(s) = π− s 2Γ( s 2)ζ(s). The function ξQ(w, s) := s(w−s) 2w ZQ(w, s) is shown to be an entire function on C×C, to satisfy the functional equation ξQ(w, s) = ξQ(w,w−s), and to have ξQ(0, s) = s2 4 (1− 2 s 2 )(1− 21− s2 )ζ̂( s 2 )ζ̂(−s 2 ). We study the location of the zeros of ZQ(w, s) for various real values of w = u. For fixed u ≥ 0 the zeros are confined to a vertical strip of width at most u+ 8 and the number of zeros Nu(T ) to height T has similar asymptotics to the Riemann zeta function. For fixed u < 0 these functions are strictly positive on the “critical line” R(s) = u2 , hence have no zeros there. This phenomenon is associated to a positive convolution semigroup with parameter u ∈ R>0, which is a semigroup of infinitely divisible probability distributions. having densities Pu(x)dx for real x, where Pu(x) = 1 2πθ(1) ZQ(−u,−u/2 + ix/2), and θ(1) = π1/4/Γ(3/4). AMS Subject Classification (2000): 11M41 (Primary) 14G40, 60E07 (Secondary)