Proceedings of the Japan Academy, Series A, Mathematical Sciences

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

Yoichi Motohashi

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In the present and the next notes of this series, we shall try to illuminate a geometric structure behind the interactions that have recently been observed between mean values of zeta-functions and automorphic representations. Our discussion is hoped to be a precursor of a unified theory of mean values of automorphic $L$-functions that we are going to forge. In this note we shall deal with the spectral structure over the modular group. In the next note the Picard group will be treated, as a typical case in the complex situation. We stress that we have been inspired by the work [2] due to Cogdell and Pyatetskii-Shapiro.

Article information

Proc. Japan Acad. Ser. A Math. Sci., Volume 78, Number 3 (2002), 36-41.

First available in Project Euclid: 23 May 2006

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Zentralblatt MATH identifier

Primary: 11M06: $\zeta (s)$ and $L(s, \chi)$
Secondary: 11F70: Representation-theoretic methods; automorphic representations over local and global fields

Mean values of zeta-functions local functional equations of Jacquet-Langlands Gamma functions of representations Bessel functions of representations


Motohashi, Yoichi. A note on the mean value of the zeta and $L$-functions. XII. Proc. Japan Acad. Ser. A Math. Sci. 78 (2002), no. 3, 36--41. doi:10.3792/pjaa.78.36.

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