Pseudodifferential arithmetic and a failed attempt on the Riemann hypothesis

TL;DR

This paper explores a criterion linked to the Riemann hypothesis involving hermitian forms and Weyl calculus, but ultimately finds the approach does not yield new insights into the conjecture.

Contribution

It proposes a reduction of the Riemann hypothesis to an algebraic problem via hermitian forms and Weyl calculus, highlighting limitations of this method.

Findings

The approach does not lead to a new method of analysis.

A potential need for better integration of analysis and arithmetic.

Hints for future research directions are provided.

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.