The sum of like powers of the zeros of the Riemann zeta function

Abstract:

In this paper we discuss a method of evaluating the sum ${\sigma _r} = \sum {{\rho ^{ - r}}}$ where r is an integer greater than 1 and the sum is taken over all the complex zeros of $\zeta (s)$, the Riemann zeta function. The method requires the coefficients of the Maclaurin expansion of the entire function $f(s) = (s - 1)\zeta (s)$. These are obtained from a limit theorem of Sitaramachandrarao by the use of the Euler-Maclaurin summation formula. The sum ${\sigma _r}$ is then obtained from the logarithmic derivative of the function $f(s)$. A table of ${\sigma _r}$ is given to 30 decimals for $r = 2(1)26$.