Pseudodifferential arithmetic, Riemann and Lindel\"of hypotheses

TL;DR

This paper introduces pseudodifferential arithmetic to explicitly construct an operator linked to the Riemann zeta zeros, leading to a disproof of the conjecture about their distribution and a proof of the Lindelöf hypothesis.

Contribution

It develops pseudodifferential arithmetic to explicitly analyze operators related to the Riemann zeta zeros, challenging existing conjectures.

Findings

Disproves the conjecture that the zeros' real parts have measure less than 0.5

Shows the set of zeros' real parts has measure at least 0.5

Provides a proof of the Lindelöf hypothesis

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 has measure at least 0.5. A similar method leads to a proof of the Lindel\\"of hypothesis.