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Structural elucidation of the mean square of the Hurwitz zeta-function

✍ Scribed by Shigeru Kanemitsu; Yoshio Tanigawa; Masami Yoshimoto

Publisher
Elsevier Science
Year
2006
Tongue
English
Weight
156 KB
Volume
120
Category
Article
ISSN
0022-314X

No coin nor oath required. For personal study only.

✦ Synopsis

In this note we shall give a complete structural description of the mean square of the Hurwitz zetafunction whose study was started 50 years ago. Instead of appealing to Atkinson's dissection, we incorporate the built-in structure of the Hurwitz zeta-function as the solution of the difference equation. First we shall prove a Katsurada-Matsumoto formula from which the best asymptotic expansion for the mean square at 1 2 + it follows by K-times integration by parts, and then we shall show that their fundamental formula is essentially the N -times integration by parts of the same formula. The key is to introduce a suitable function f κ the integration of which gives 2 1 ΢(u, x)΢(v, x) dx, and then to view 2 1 f κ (x; u, v) dx as ∞ 1f κ (x; u, v) dx.

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