Explicit formula on function fields and application: Li coefficients
From the text: ``\textit{X.-J. Li} [J. Number Theory 65, No. 2, 325-333 (1997; Zbl 0884.11036)] derived a necessary and sufficient condition for the Riemann hypothesis in terms of the positivity of the set of coefficients \[ \lambda_n=\sum\limits_\rho {}^*\left[ 1-\left( 1-\frac1\rho\right)^n\right] \] for all positive integers \(n\), where the sum is taken over all the non-trivial zeros of the Riemann zeta function and \(\ast\) denotes that the sum is taken in the sense of the limit \(\lim\limits_{|\Im(\rho)|\to \infty}\)''. ``In this note, we formulate an explicit formula for the completed zeta function of a function field \(K\) of genus \(g\) over a finite field \(\mathbb F_q\), analogous to the Weil explicit formula. Then we give an arithmetic formula for the nth Li coefficient \(\lambda_K(n)\) for the function field \(K\). Furthermore, as an application, we give some formulas for the Euler-Stieltjes constants \(\gamma_K\) on function fields.''
scientific article; zbMATH DE number 7374202
scientific article