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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

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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Additional Information

DOI: http://dx.doi.org/10.1090/S0025-5718-1988-0917834-X
PII: S 0025-5718(1988)0917834-X
Article copyright: © Copyright 1988 American Mathematical Society