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Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.78.

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The sum of like powers of the zeros of the Riemann zeta function
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by D. H. Lehmer PDF
Math. Comp. 50 (1988), 265-273 Request permission

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
  • © Copyright 1988 American Mathematical Society
  • Journal: Math. Comp. 50 (1988), 265-273
  • MSC: Primary 11M26; Secondary 11Y35
  • DOI: https://doi.org/10.1090/S0025-5718-1988-0917834-X
  • MathSciNet review: 917834