Variants of the Li-type criteria for the generalized Riemann hypothesis (Q6611639)

This paper under review is a nice survey of the Li-type criteria for the (GRH) and the \(\tau/2\)-(GRH) which have generated a lot of interest in the past 25 years. The authors present an overview of results related to the analytic and numerical study of the generalized Li-type coefficients that are present in the vast literature on this topic. For instance, different expressions for the Li and \(\tau\)-Li coefficients are presented relating the (generalized) Li coefficients with the prime number theorem, the Euler-Stieltjes constants, and other arithmetic data. The \(\tau\)-Li coefficients have also been related to explicit zero-free regions of \(L\)-functions. Furthermore, the authors focus on various expressions of the generalized Li and \(\tau\)-Li coefficients which are suitable for numerical experiments in order to establish asymptotic behavior of those coefficients. Finally, the paper concludes with a table summarizing the bibliography related to the Li coefficients and the Li-type criteria for different classes of \(L\)-functions.\N\NFor the entire collection see [Zbl 1542.11001].