Searcharxiv⌕ Search

arXiv subjects

André Voros

Publications and source records attributed to André Voros.

11 recordsLinked to original sources
arXiv2022

From asymptotic to closed forms for the Keiper/Li approach to the Riemann Hypothesis

The Riemann Hypothesis (RH) - that all nonreal zeros of Riemann's zeta function shall have real part 1/2 - remains a major open problem. Its most concrete equivalent is that an infinite sequence of real numbers, the Keiper--Li constants, shall be everywhere positive (Li's criterion). But those numbers are analytically elusive and strenuous to compute, hence we seek simpler variants. The essential sensitivity to RH of that sequence lies in its asymptotic tail; then, retaining this feature, we can modify the Keiper--Li scheme to obtain a new sequence in elementary closed form. This makes for a more explicit analysis, with easier and faster computations. We can moreover show how the new sequence will signal RH-violating zeros if any, by observing its analogs for the Davenport--Heilbronn counterexamples to RH.

math.NT↗
arXiv2022

Discretized Keiper/Li approach to the Riemann Hypothesis

The Keiper--Li sequence $\{ λ_n \}$ is most sensitive to the Riemann Hypothesis asymptotically ($n \to \infty$), but highly elusive both analytically and numerically. We deform it to fully explicit sequences, simpler to analyze and to compute (up to $n=5 \cdot 10^5$ by G. Misguich). This also works on the Davenport--Heilbronn counterexamples, thus we can demonstrate explicit tests that selectively react to zeros off the critical line.

math.NT↗
arXiv2016

Simplifications of the Keiper/Li approach to the Riemann Hypothesis

The Keiper/Li constants $\{λ_n\}_{n=1,2,\ldots}$ are asymptotically ($n \to \infty$) sensitive to the Riemann Hypothesis, but highly elusive analytically and difficult to compute numerically. We present quite explicit variant sequences that stay within the abstract Keiper--Li frame, and appear simpler to analyze and compute.

math.NT↗
arXiv2014

Zeta functions over zeros of Zeta functions and an exponential-asymptotic view of the Riemann Hypothesis

We review generalized zeta functions built over the Riemann zeros (in short: "superzeta" functions). They are symmetric functions of the zeros that display a wealth of explicit properties, fully matching the much more elementary Hurwitz zeta function. As a concrete application, a superzeta function enters an integral representation for the Keiper--Li coefficients, whose large-order behavior thereby becomes computable by the method of steepest descents; then the dominant saddle-point entirely depends on the Riemann Hypothesis being true or not, and the outcome is a sharp exponential-asymptotic criterion for the Riemann Hypothesis that only refers to the large-order Keiper--Li coefficients. As a new result, that criterion, then Li's criterion, are transposed to a novel sequence of Riemann-zeta expansion coefficients based at the point 1/2 (vs 1 for Keiper--Li).

math.NT↗
arXiv2012

Zeta-regularisation for exact-WKB resolution of a general 1D Schrödinger equation

We review an exact analytical resolution method for general one-dimensional (1D) quantal anharmonic oscillators: stationary Schrödinger equations with polynomial potentials. It is an exact form of WKB treatment involving spectral (usual) vs "classical" (newer) zeta-regularisations in parallel. The central results are a set of Bohr--Sommerfeld-like but exact quantisation conditions, directly drawn from Wronskian identities, and appearing to extend Bethe-Ansatz formulae of integrable systems. Such exact quantisation conditions do not just select the eigenvalues; some evaluate the spectral determinants, and others the wavefunctions, for the spectral parameter in general position.

math-ph↗
arXiv2006

Sharpenings of Li's criterion for the Riemann Hypothesis

Exact and asymptotic formulae are displayed for the coefficients $λ_n$ used in Li's criterion for the Riemann Hypothesis. For $n \to \infty$ we obtain that if (and only if) the Hypothesis is true, $λ_n \sim n(A \log n +B)$ (with $A>0$ and $B$ explicitly given, also for the case of more general zeta or $L$-functions); whereas in the opposite case, $λ_n$ has a non-tempered oscillatory form.

math.NT↗
arXiv2004

The general 1D Schrödinger equation as an exactly solvable problem

We review an exact WKB resolution method for the stationary 1D Schrödinger equation with a general polynomial potential. This contribution covers already published material: we supply a commented summary here, stressing a few aspects which were less highlighted before. As some of our earlier papers needed later corrections, we also recapitulate these here in footnotes in the bibliography. The latter, not meant to be exhaustive, focuses on narrowly relevant works plus some broader ones that do offer more extensive bibliographies.

math-ph↗
arXiv2004

A sharpening of Li's criterion for the Riemann Hypothesis

Exact and asymptotic formulae are displayed for the coefficients $λ_n$ used in Li's criterion for the Riemann Hypothesis. In particular, we argue that if (and only if) the Hypothesis is true, $λ_n \sim n(A \log n +B)$ for $n \to \infty$ (with explicit $A>0$ and $B$). The approach also holds for more general zeta or $L$-functions.

math.NT↗
arXiv2003

From exact-WKB towards singular quantum perturbation theory

We use exact WKB analysis to derive some concrete formulae in singular quantum perturbation theory, for Schrödinger eigenvalue problems on the real line with polynomial potentials of the form $(q^M + g q^N)$, where $N>M>0$ even, and $g>0$. Mainly, we establish the $g \to 0$ limiting forms of global spectral functions such as the zeta-regularized determinants and some spectral zeta functions.

math-ph↗