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

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

Yoichi Motohashi

#### Abstract

The aim of the present article is to render the spectral theory of mean values of automorphic $L$-functions -- in a unified fashion. This is an outcome of the investigation commenced with the parts XII and XIV, where a framework was laid on the basis of the theory of automorphic representations and a general approach to the mean values was envisaged. We restrict ourselves to the situation offered by the full modular group, solely for the sake of simplicity. Details and extensions are to be published elsewhere.

#### Article information

Source
Proc. Japan Acad. Ser. A Math. Sci., Volume 83, Number 6 (2007), 73-78.

Dates
First available in Project Euclid: 29 August 2007

https://projecteuclid.org/euclid.pja/1188405574

Digital Object Identifier
doi:10.3792/pjaa.83.73

Mathematical Reviews number (MathSciNet)
MR2355501

Zentralblatt MATH identifier
1196.11073

#### Citation

Motohashi, Yoichi. A note on the mean value of the zeta and $L$-functions. XV. Proc. Japan Acad. Ser. A Math. Sci. 83 (2007), no. 6, 73--78. doi:10.3792/pjaa.83.73. https://projecteuclid.org/euclid.pja/1188405574

#### References

• V. Blomer and G. Harcos, The spectral decomposition of shifted convolution sums. arXiv: math/0703246.
• R. W. Bruggeman and Y. Motohashi, A new approach to the spectral theory of the fourth moment of the Riemann zeta-function, J. Reine Angew. Math. 579 (2005), 75–114.
• Y. Motohashi, The mean square of Hecke $L$-series attached to holomorphic cusp-forms, Sūrikaisekikenkyūsho Kōkyūroku No.886 (1994), 214–227.
• Y. Motohashi, Spectral theory of the Riemann zeta-function, Cambridge Univ. Press, Cambridge, 1997.
• Y. Motohashi, 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.
• Y. Motohashi, 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.
• Y. Motohashi, Mean values of zeta-functions via representation theory, in Multiple Dirichlet series, automorphic forms, and analytic number theory, 257–279, Proc. Sympos. Pure Math., 75, Amer. Math. Soc., Providence, RI, 2006.