On the Chowla–Selberg integral formula for non-holomorphic Eisenstein series

Abstract

In this note, we get the Fourier expansion for the non-holomorphic Eisenstein series by slight modification of Maass’ original method, which enables us to prove as a bonus, two integral representations for the modified Bessel function of the third kind. This kind reveals a hidden inner structure of the non-holomorphic Eisenstein series and the Bessel diffenrential equation. We also explain a work of Motohashi on the Kronecker limit formula for the Epstein zeta-function from our point of view.

Keywords:

• Chowla–Selberg integral formula
• non-holomorphic Eisenstein series
• Bessel function

1991 Mathematics Subject Classification :

• Primary: 11F66
• 11M26
• Secondary: 33B15

Acknowledgements

The author thanks Prof. S. Kanemitsu for suggesting the problem and for many discussions that he had with the latter during Prof. Kanemitsu's recent visit to Harish-Chandra Research Institute. The author also thanks Prof. J.-Y. Liu for making available his lecture notes 8 on ‘Maass forms’ and Prof. H.M. Srivastava for his kind suggestions which have helped in improving the presentation of this paper.