arXiv subjects

André Unterberger

Publications and source records attributed to André Unterberger.

3 recordsLinked to original sources

arXiv2026

Pseudodifferential arithmetic, Riemann and Lindelöf hypotheses

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: zeros accumulate on the boundary of the critical strip. A similar method leads to a proof of the Lindel\öf hypothesis.

math.NT↗

arXiv2024

Pseudodifferential arithmetic and the Riemann hypothesis: reminders

The present preprint completes the arXiv preprint #2202.11652, entitled "Pseudodifferential arithmetic and the Riemann hypothesis", devoted to a proof of the conjecture. The first 4 pages of that preprint were devoted to a set of necessary reminders, given in a very concise way: we here give a self-contained, fully developed, version of this part.

math.NT↗

arXiv2006

Projective Pseudodifferential Analysis and Harmonic Analysis

We consider pseudodifferential operators on functions on $\R^{n+1}$ which commute with the Euler operator, and can thus be restricted to spaces of functions homogeneous of some given degree. Their symbols can be regarded as functions on a reduced phase space, isomorphic to the homogeneous space $G_n/H_n=SL(n+1,\R)/GL(n,\R)$, and the resulting calculus is a pseudodifferential analysis of operators acting on spaces of appropriate sections of line bundles over the projective space $P_n(\R)$ : these spaces are the representation spaces of the maximal degenerate series $(π_{iλ,ε})$ of $G_n$ . This new approach to the quantization of $G_n/H_n$, already considered by other authors, has several advantages: as an example, it makes it possible to give a very explicit version of the continuous part from the decomposition of $L^2(G_n/H_n)$ under the quasiregular action of $G_n$ . We also consider interesting special symbols, which arise from the consideration of the resolvents of certain infinitesimal operators of the representation $π_{iλ,ε}$ .

math.RT↗