Proceedings of the Japan Academy, Series A, Mathematical Sciences

A note on the mean value of the zeta and $L$-functions. X

Roelof Wichert Bruggeman and Yoichi Motohashi

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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.

Article information

Proc. Japan Acad. Ser. A Math. Sci., Volume 77, Number 7 (2001), 111-114.

First available in Project Euclid: 23 May 2006

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11M06: $\zeta (s)$ and $L(s, \chi)$
Secondary: 11F72: Spectral theory; Selberg trace formula

Zeta-function imaginary quadratic number field Kloosterman sum sum formula automorphic representation spectral decomposition


Bruggeman, Roelof Wichert; Motohashi, Yoichi. A note on the mean value of the zeta and $L$-functions. X. Proc. Japan Acad. Ser. A Math. Sci. 77 (2001), no. 7, 111--114. doi:10.3792/pjaa.77.111.

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