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A logarithm type mean value theorem of the Riemann zeta function

✍ Scribed by Xia-Qi Ding; Shao-Ji Feng

Publisher: Elsevier Science
Year: 2006
Tongue: English
Weight: 90 KB
Volume: 120
Category: Article
ISSN: 0022-314X
DOI: 10.1016/j.jnt.2005.12.007

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✦ Synopsis

For any integer K 2 and positive integer h, we investigate the mean value of |ζ(σ + it)| 2k × log h |ζ(σ + it)| for all real number 0 < k < K and all σ > 1 -1/K. In case K = 2, h = 1, this has been studied by Wang in [F.T. Wang, A mean value theorem of the Riemann zeta function, Quart. J. Math. Oxford Ser. 18 (1947) 1-3]. In this note, we give a new brief proof of Wang's theorem, and, with this method, generalize it to the general case naturally.

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