Abstract
Pseudodifferential analysis helps linking the one-dimensional and two-dimensional analyses in more than one way. Taking benefit from special features of the two-dimensional case, in particular the fact that homogeneous distributions, invariant under the action by linear changes of coordinates of \(SL(2,{\mathbb Z})\), are essentially disguised versions of nonholomorphic modular forms, we are led to introducing interesting two-dimensional distributions and some associated one-dimensional parts. This results in a collection of necessary and sufficient conditions for the Riemann hypothesis to hold, some of which, but not all, are of a more or less classical type.
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