Evaluation of the Li coefficients on function fields and applications
The classical Riemann hypothesis is equivalent to several statements. In particular, it is equivalent to $\lambda_n\geq 0$ for $n\geq 1,$ where $\lambda_n:= \sum_{\rho}^*\big[1-\big(1-\frac{1}{\rho}\big)^n\big]$. The sum runs through the nontrivial zeros of the Riemann zeta function and $^*$ denotes that the sum is taken in the sense of the limit $\lim_{|(\rho)|\to\infty}$. The Riemann hypothesis for global function fields was proved by A. Weil in 1940--1941. The $n$-th Li coefficient for a global function field $K$ is defined as \[ \lambda_K(n)= \mathop{{\sum}^*}_{\rho \in {\mathcal Z}(K)} \big(1-\big(1-\tfrac{1}{\rho}\big)^n\big)=\lim_{T\to \infty}\sum_{\substack{ {\rho\in{\mathcal Z}(K)}\\ |\mathrm{Im}(\rho)|\leq T}} \big(1-\big(1-\tfrac{1}{\rho}\big)^n\big)\] where ${\mathcal Z}(K)$ is the set of zeros of the Riemann zeta function $\zeta_K(s)$. We have that the Riemann hypothesis is equivalent to $\lambda_K(n)\geq 0$ for $n\geq 1$. In the paper under review, the authors first derive several formulas to compute $\lambda_K(n)$ (Theorems 3.2, 4.1, 4.2 and Corollary 3.3). Second, they apply the above formulas and the inequalities $\lambda_K(n)\geq 0$ to approximate some arithmetic invariants of the field $K$ (Corollaries 5.1, 5.2, 5.3 and Example 5.4). For instance, they show that \[ g h_K\max\big\{1,2\big(1-\frac{2}{\log q}\big) \big\}\leq L'(1)\leq 2 gh_K\big(1+\frac{2}{\log q}\big)\] where $L$ is the $L$--function associated to $K$, $g$ is the genus and $h_K$ is the class number of $K$.
scientific article; zbMATH DE number 7073696
scientific article