Pseudodifferential arithmetic, Riemann and Lindelöf hypotheses

Abstract

The Weyl symbolic calculus of operators leads to the construction, if one takes for symbol a certain distribution decomposing over the zeros of the Riemann zeta function, of an operator with the following property: the Riemann hypothesis is equivalent to the validity of a collection of estimates involving this operator. Pseudodifferential arithmetic, a novel chapter of pseudodifferential operator theory, makes it possible to make the operator under study fully explicit. This leads to a disproof of the conjecture: the closure of the set of real parts of non-trivial zeros of zeta is dense in $(0,1)$. A similar method leads to a proof of the Lindel\öf hypothesis.