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Mathematics of Computation

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

Author: D. H. Lehmer
Journal: Math. Comp. 50 (1988), 265-273
MSC: Primary 11M26; Secondary 11Y35
MathSciNet review: 917834
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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 $r = 2(1)26$.

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Article copyright: © Copyright 1988 American Mathematical Society