Li's criterion and the Riemann hypothesis for function fields (Q609365)

Let \(K\) be a function field of genus \(g\) over a finite field \({\mathbb F}_q\) with zeta function \(\zeta_K(s)\) and Li coefficients \(\lambda_K(n)\), \(n\in\mathbb N\). The authors prove that the zeros of \(\zeta_K(s)\) lie on the line \(\text{Re}(s)=\frac12\), if and only if \[ |\lambda_K(n)|\leq 2gq^{n/2},\quad\forall n\in {\mathbb N}. \] It holds that \[ \lambda_K(n)=2(g-1)\log qn+o(n), \quad n\to\infty. \] They show, in particular, that the Riemann hypothesis for the function field \(K\) holds if and only if \( | N_n-(q^n+1)|\leq 2gq^{n/2}\) 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 \(K\).