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Vanishing of the integral of the Hurwitz zeta Function

Published online by Cambridge University Press:  17 April 2009

Kevin A. Broughan
Kevin A. Broughan
Affiliation:
Department of Mathematics, University of Waikato, Hamilton, New Zealand e-mail: kab@waikato.ac.nz
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Abstract

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A proof is given that the improper Riemann integral of ζ(s, a) with respect to the real parameter a, taken over the interval (0, 1], vanishes for all complex s with ℜ(s) < 1. The integral does not exist (as a finite real number) when ℜ(s) ≥ 1.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 65 , Issue 1 , February 2002 , pp. 121 - 127
Copyright
Copyright © Australian Mathematical Society 2002

References

[1]Apostol, T.M., Introduction to analytic number theory (Springer-Verlag, New York, Berlin, Heidelberg, 1976).Google Scholar
[2]Hansen, E.R. and Patrick, M.L., ‘Some relations and values for the generalized Riemann zeta function’, Math. Comp. 16 (1962), 265–274.Google Scholar
[3]Katsurada, M. and Matsumoto, K., ‘Discrete values of Hurwitz zeta-function’, Proc. Japan Acad. Ser. A Math. Sci. 69 (1993), 164–169.CrossRefGoogle Scholar
[4]Kumchev, A., ‘A note on the 2k-the mean value of the Hurwitz zeta function’, Bull. Austral. Math. Soc. 60 (1999), 403–405.CrossRefGoogle Scholar
[5]Rane, V.V., ‘A new approximate functional equation for Hurwitz zeta function for rational parameter’, Proc. Indian Acad. Sci. Math. Sci. 107 (1997), 377–385.Google Scholar
[6]Zhang, W., ‘On the forth power mean of Hurwitz zeta-function’, Pure Appl. Math. 12 (1996), 13–18.Google Scholar
[7]Zhang, W., ‘On the mean square value of the Hurwitz zeta-function’, Illinois J. Math. 38 (1994), 71–78.Google Scholar