Beginning with Wiener's paper, ``Tauberian Theorems'' in the Annals of Math (1932) a number of functional analytic equivalences of RH have been proven. These involve the completeness of various spaces. M. Balazard has recently written an excellent survey on these developments (See Surveys in Number Theory, Papers from the Millenial Conference on Number Theory, A. K. Peters, 2003.)
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