The two-variable zeta function and the Riemann hypothesis for function fields

Riemann Hypothesis for function fields

1 1 Preliminaries 1 1.1 Function fields . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 The zeta function . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2.1 Primes and Divisors . . . . . . . . . . . . . . . . . . . . 2 1.2.2 The Picard Group . . . . . . . . . . . . . . . . . . . . . . 5 1.2.3 Riemann-Roch . . . . . . . . . . . . . . . . . . . . . . . 6 1.3 Notation . . . . ...

On the Riemann Hypothesis for function fields

We prove a variant of Connes’s trace formula and show how it can be used to give a new proof of the Riemann hypothesis for L-functions with Größencharacter for function fields.

On The Riemann Hypothesis For The Characteristic p Zeta Function

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 variable...

Mollifying the Riemann Zeta-function