An extension of Maass theory to general Dirichlet series

ABSTRACT

In this paper we establish the modular relation for Maass forms to the effect that the Fourier–Whittaker expansion and the ramified functional equation are equivalent, i.e. the RHB (Riemann–Hecke–Bochner) correspondence. This arises from the new standpoint that Maass' procedure is one of the ways leading to the Fourier–Whittaker expansion similar to the situation that various representations of the modified Bessel function $K_{\nu }$ have provided such vast amount of different methods including the Hardy transform and the beta-transform leading to the Fourier-Bessel expansion from opposite directions. Naturally, we may abridge the gap between Hecke Eisenstein series and Epstein zeta-functions and the Chowla–Selberg integral formula and Ramanujan–Guinand formula are immediate corollaries, whose remark is due to the referee.

KEYWORDS:

• Beta-transform
• Fourier–Whittaker expansion
• Fourier–Bessel expansion
• Maass form
• Hardy transform

AMS CLASSIFICATIONS:

• 11B83
• 11A25
• 11E45

Acknowledgments

We would like to express our hearty thanks to the referees for their enlightening reviews. We are very grateful to one of them who kindly wrote three research paper-equivalent reports of 16, 6 and 5 pages! It was so thorough and penetrating that we felt awed. Because of limitation of time and space, we could not incorporate all of the referee's suggestions nor answers to questions remain partial, and we will surely keep them as a guiding principle for further research. The early manuscript has been totally rewritten incorporating the referee's suggestions. In the second report the referee further mentioned two more papers which were not known to the authors, [48,49]. We would like to turn to the study of these and other points suggested by the referee elsewhere.

Disclosure statement

No potential conflict of interest was reported by the author(s).