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 σ r = ∑ ρ − r {\sigma _r} = \sum {{\rho ^{ - r}}} where r is an integer greater than 1 and the sum is taken over all the complex zeros of ζ ( s ) \zeta (s) , the Riemann zeta function. The method requires the coefficients of the Maclaurin expansion of the entire function f ( s ) = ( s − 1 ) ζ ( s ) 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 σ r {\sigma _r} is then obtained from the logarithmic derivative of the function f ( s ) f(s) . A table of σ r {\sigma _r} is given to 30 decimals for r = 2 ( 1 ) 26 r = 2(1)26 .