In 1949 Hans Maass made his profound discovery that there are L-functions associated with non-holomorphic automorphic forms. In his Math Review of Maass' article [MR:0031519] J. Lehner writes

*In Hecke's theory of Dirichlet series with Euler products we associate, roughly speaking,
a Dirichlet series with an automorphic function; the invariance of the latter under linear
substitutions is used, together with the Mellin transform, to derive a functional equation
for the Dirichlet series. This suffices for the discussion of the $\zeta$-function of an imaginary
quadratic field, for example, but not of a real quadratic field. In order to handle the latter case,
the author defines a class of functions
("automorphic wave functions") to take the place of the analytic
automorphic functions of Hecke's theory. *

Maass' work prompted André Weil to remark “Il a fallu Maass pour nous sortir du ghetto des fonctions holomorphes.”

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- Last edited by Brian Conrey on 2016-05-10 01:56:31

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- 2016-05-10 01:56:31 by Brian Conrey (Reviewed)