Computation of the secondary zeta function

Abstract

The secondary zeta function $Z(s)={\sum }_{n=1}^{\infty }{\alpha }_n^{-s}$, where ${\rho }_n=\frac{1}{2}+i{\alpha }_n$ are the zeros of zeta with $\Im (\rho )>0$, extends to a meromorphic function on the hole complex plane. If we assume the Riemann hypothesis the numbers ${\alpha }_n={\gamma }_n$, but we do not assume the RH. We give an algorithm to compute the analytic prolongation of the Dirichlet series $Z(s)={\sum }_{n=1}^{\infty }{\alpha }_n^{-s}$, for all values of $s$ and to a given precision.

19 pages, 11 figures

References in corpus (1)

• On certain sums over ordinates of zeta zeros