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Sept. 2001 A note on the mean value of the zeta and $L$-functions. X
Roelof Wichert Bruggeman, Yoichi Motohashi
Proc. Japan Acad. Ser. A Math. Sci. 77(7): 111-114 (Sept. 2001). DOI: 10.3792/pjaa.77.111


The present note reports on an explicit spectral formula for the fourth moment of the Dedekind zeta function $\zeta_{\mathrm{F}}$ of the Gaussian number field $\mathrm{F} = \mathbf{Q}(i)$, and on a new version of the sum formula of Kuznetsov type for $\mathrm{PSL}_2(\mathbf{Z}[i])\backslash \mathrm{PSL}_2(\mathbf{C})$. Our explicit formula (Theorem 5, below) for $\zeta_{\mathrm{F}}$ gives rise to a solution to a problem that has been posed on p. 183 of [M3] and, more explicitly, in [M4]. Also, our sum formula (Theorem 4, below) is an answer to a problem raised in [M4] concerning the inversion of a spectral sum formula over the Picard group $\mathrm{PSL}_2(\mathbf{Z}[i])$ acting on the three dimensional hyperbolic space (the $K$-trivial situation). To solve this problem, it was necessary to include the $K$-nontrivial situation into consideration, which is analogous to what has been experienced in the modular case.


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Roelof Wichert Bruggeman. Yoichi Motohashi. "A note on the mean value of the zeta and $L$-functions. X." Proc. Japan Acad. Ser. A Math. Sci. 77 (7) 111 - 114, Sept. 2001.


Published: Sept. 2001
First available in Project Euclid: 23 May 2006

zbMATH: 1049.11124
MathSciNet: MR1857285
Digital Object Identifier: 10.3792/pjaa.77.111

Primary: 11M06
Secondary: 11F72

Keywords: automorphic representation , imaginary quadratic number field , Kloosterman sum , spectral decomposition , sum formula , zeta-function

Rights: Copyright © 2001 The Japan Academy

Vol.77 • No. 7 • Sept. 2001
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