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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Mathematics of Computation is 1.98.

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Values of the Legendre chi and Hurwitz zeta functions at rational arguments
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by Djurdje Cvijović and Jacek Klinowski PDF
Math. Comp. 68 (1999), 1623-1630 Request permission

Abstract:

We show that the Hurwitz zeta function, $\zeta (\nu ,a)$, and the Legendre chi function, $\chi _\nu (z)$, defined by \[ \zeta (\nu ,a)=\sum _{k=0}^\infty \frac {1}{(k+a)^\nu },\quad 0<a\le 1,\operatorname {Re} \nu >1,\] and \[ \chi _\nu (z)=\sum _{k=0}^\infty \frac {z^{2k+1}}{(2k+1)^\nu },\quad |z|\le 1,\operatorname {Re} \nu >1 \text {with} \nu =2,3,4,\dotsc ,\] respectively, form a discrete Fourier transform pair. Many formulae involving the values of these functions at rational arguments, most of them unknown, are obtained as a corollary to this result. Among them is the further simplification of the summation formulae from our earlier work on closed form summation of some trigonometric series for rational arguments. Also, these transform relations make it likely that other results can be easily recovered and unified in a more general context.
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Additional Information
  • Djurdje Cvijović
  • Affiliation: Department of Chemistry, University of Cambridge, Lensfield Road, Cambridge CB2 1EW, United Kingdom
  • Email: dc133@cam.ac.uk
  • Jacek Klinowski
  • Affiliation: Department of Chemistry, University of Cambridge, Lensfield Road, Cambridge CB2 1EW, United Kingdom
  • Email: jk18@cam.ac.uk
  • Received by editor(s): February 16, 1998
  • Published electronically: May 17, 1999
  • © Copyright 1999 American Mathematical Society
  • Journal: Math. Comp. 68 (1999), 1623-1630
  • MSC (1991): Primary 65B10; Secondary 11M35
  • DOI: https://doi.org/10.1090/S0025-5718-99-01091-1
  • MathSciNet review: 1648375