Zeta functions over zeros of zeta functions
Let \(\rho\) be a non-real zero of the Riemann zeta function \(\zeta(s)\). Then it can be written as \(\rho=\frac12\pm i\tau\) with \(\text{Re}(\tau)>0\) and thus \(\rho(1-\rho)=\frac14+\tau^2\). The present book deals with three types of sums which come about by summing over these zeros \(\rho\) and which are called superzeta functions. The first kind is \[ \sum_\rho(z-\rho)^{-s},\qquad \text{Re}(s)>1, \] where the sum runs over all \(\rho\), each repeated according to its multiplicity. Here one defines \((z-\rho)^{-s}=\exp(-s\log(z-\rho))\), where the principal branch of the logarithm is taken. To avoid points of discontinuity or undefined points of the logarithm, one restricts the shift parameter \(z\) in a way that for every \(\rho\) the number \(z-\rho\) lies in the cut plane \({\mathbb C}\smallsetminus (-\infty,0]\). Under similar restrictions one defines the second kind, \[ \sum_\tau(z+\tau^2)^{-s},\qquad \text{Re}(s)>\frac12, \] and the third, \[ \sum_{\tau}(z+\tau)^{-s},\qquad\text{Re}(s)>1. \] In all three cases the main questions are for analytic continuation and the analytic properties of the ensuing functions. Note that the third kind is the only one for which the symmetry \(\rho\mapsto (1-\rho)\) is broken. So it is not surprising, that the function of the third kind is analytically less well behaved than the previous two. The main results of the book are explicit expressions in either case from which the meromorphic continuation (in \(s\)) can be read off. Also, various special values are given. The results are then extended to more general Dirichlet-series rather than \(\zeta(s)\) only, but here technical restrictions are to be imposed which leave only functions pretty near to \(\zeta(s)\). As an application, a new criterion for the truth of the Riemann hypothesis is given. The main chapters of the book need a lot of technical preparation, consisting of known facts from the literature, which are collected in the first four chapters of the book. It follows an introduction to superzeta functions, then a careful analysis of the three types, an extension to more general Dirichlet-series, and the application, a new criterion for the Riemann hypothesis. The book is very carefully written and presents an object of research which has been investigated for many decades but never treated in such a comprehensive manner. A valuable contribution to the field of analytic number theory.
scientific article; zbMATH DE number 5633958