## Proceedings of the Japan Academy, Series A, Mathematical Sciences

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

Yoichi Motohashi

#### Abstract

The aim of the present note is to develop a study on the feasibility of a unified theory of mean values of automorphic $L$-functions, a desideratum in the field. This is an outcome of the investigation commenced with the part XII ([14]), where a framework was laid on the basis of the theory of automorphic representations, and a general approach to the mean values was envisaged. Specifically, it is shown here that the inner-product method, which was initiated by A. Good [7] and greatly enhanced by M. Jutila [9], ought to be brought to perfection so that the mean square of the $L$-function attached to any cusp form on the upper half-plane could be reached within the notion of automorphy. The Kirillov map is our key implement. Because of its geometric nature, our method appears to extend to bigger linear Lie groups. This note is essentially self-contained.

#### Article information

Source
Proc. Japan Acad. Ser. A Math. Sci., Volume 80, Number 4 (2004), 28-33.

Dates
First available in Project Euclid: 18 May 2005

https://projecteuclid.org/euclid.pja/1116442213

Digital Object Identifier
doi:10.3792/pjaa.80.28

Mathematical Reviews number (MathSciNet)
MR2055073

Zentralblatt MATH identifier
1052.11035

#### Citation

Motohashi, Yoichi. A note on the mean value of the zeta and $L$-functions. XIV. Proc. Japan Acad. Ser. A Math. Sci. 80 (2004), no. 4, 28--33. doi:10.3792/pjaa.80.28. https://projecteuclid.org/euclid.pja/1116442213

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