More zeta functions for the Riemann zeros. 2nd printing. (English) Zbl 1147.11048

Cartier, Pierre (ed.) et al., Frontiers in number theory, physics, and geometry I. On random matrices, zeta functions, and dynamical systems. Papers from the meeting, Les Houches, France, March 9–21, 2003. Berlin: Springer (ISBN 978-3-540-23189-9/hbk). 349-363 (2006).
As the title indicates and the author explains, this paper is a partial expansion of an homonymous work of the author, where zeta functions built over the non-trivial zeros of the Riemann zeta-function were studied. In the new paper, another family of generalized zeta functions built over the Riemann zeros, \(\{ \rho \}\), is considered, namely \({\mathcal Z} (s,x)= \sum_\rho (x-\rho )^{-s}\).
After a summary of the previous results, in particular on zeta functions and their use for regularizing infinite products, analytical properties of the new zeta function are determined and countably many special values of this function are listed in explicit detail. The final remark is that the analysis in the paper can be extended to zeros of zeta and \(L\)-functions having functional equations similar enough to that of the Riemann zeta-function.
For the entire collection see [Zbl 1106.11002].


11M26 Nonreal zeros of \(\zeta (s)\) and \(L(s, \chi)\); Riemann and other hypotheses
11M36 Selberg zeta functions and regularized determinants; applications to spectral theory, Dirichlet series, Eisenstein series, etc. (explicit formulas)