Li's criterion and the Riemann hypothesis for function fields

In this paper, we extend Li's criterion for a function field K of genus g   over a finite field FqFq. We prove that the zeros of the zeta-function of K   lie on the line Re(s)=12 if and only if the Li coefficients λK(n)λK(n) satisfy|λK(n)|⩽2gqn/2for alln∈N. Therefore, we particularly show that the Riemann hypothesis for the function field K holds if and only if|Nn−(qn+1)|⩽2gqn/2for alln∈N, where Nn=|X(Fqn)|Nn=|X(Fqn)| is the number of FqnFqn-rational points on the curve XX associated to the function field K  . Finally, we give an explicit asymptotic formula for the Li coefficients λK(n)λK(n).