Pseudodifferential arithmetic and a failed attempt on the Riemann hypothesis

Abstract

A criterion for the validity of the Riemann hypothesis reduced the problem to the search for a certain estimate, for a hermitian form associated by means of the Weyl symbolic calculus of operators to a distribution in the plane of an arithmetic nature. One can reduce the question further to an algebraic question. After completion of the calculation of the hermitian form obtained, this attempt does not seem to lead to a genuinely new method of analysis of the conjecture. A better cooperation between usual analysis and congruence arithmetic may be called for, and some possible hints are given at the end.