Aspects of the zeta function originating from pseudodifferential analysis

摘要

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, 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.