Li's criterion and the Riemann hypothesis for function fields

Abstract

In this paper, we extend Li's criterion for a function field K of genus g over a finite field F q . We prove that the zeros of the zeta-function of K lie on the line Re ( s ) = 1 2 if and only if the Li coefficients λ K ( n ) satisfy | λ K ( n ) | ⩽ 2 g q n / 2 for all n ∈ N . Therefore, we particularly show that the Riemann hypothesis for the function field K holds if and only if | N n − ( q n + 1 ) | ⩽ 2 g q n / 2 for all n ∈ N , where N n = | X ( F q n ) | is the number of F q n -rational points on the curve X associated to the function field K. Finally, we give an explicit asymptotic formula for the Li coefficients λ K ( n ) .