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April 2004 A note on the mean value of the zeta and $L$-functions. XIV
Yoichi Motohashi
Proc. Japan Acad. Ser. A Math. Sci. 80(4): 28-33 (April 2004). DOI: 10.3792/pjaa.80.28


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.


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Yoichi Motohashi. "A note on the mean value of the zeta and $L$-functions. XIV." Proc. Japan Acad. Ser. A Math. Sci. 80 (4) 28 - 33, April 2004.


Published: April 2004
First available in Project Euclid: 18 May 2005

zbMATH: 1052.11035
MathSciNet: MR2055073
Digital Object Identifier: 10.3792/pjaa.80.28

Primary: 11F70

Keywords: automorphic representations of linear Lie groups , Kirillov map , Mean values of automorphic $L$-functions

Rights: Copyright © 2004 The Japan Academy

Vol.80 • No. 4 • April 2004
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