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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 functionhave 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.

An extension of Maass theory to general Dirichlet series