An approximate formula for the Riemann zeta function
Let 0\(\leq\sigma \leq 1\), \(t>0\), \(m=[(t/2\pi)^{1/2}]\), and \(N<At\) where A is a sufficiently small constant. Then the Riemann-Siegel formula states \[ \zeta (s)=\sum^{m}_{n=1}n^{-s}+\pi^{-1}(2\pi)^ s\quad\sin (\pi s/2)\quad\Gamma (1-s)\sum^{m}_{n=1}n^{s-1}+ \] \[ (-1)^{m- 1}\quad e^{-i\pi (s-1)/2}\quad (2\pi t)^{(s-1)/2}+E \] where E is the sum of an asymptotic series and two small error terms. In this paper the term E is generalized by introduction of some further parameters.
scientific article; zbMATH DE number 3875392