53 research outputs found
Simplifications of the Keiper/Li approach to the Riemann Hypothesis
The Keiper/Li constants are asymptotically () 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.Comment: 21 pages, 6 figure
From exact-WKB towards singular quantum perturbation theory
We use exact WKB analysis to derive some concrete formulae in singular
quantum perturbation theory, for Schr\"odinger eigenvalue problems on the real
line with polynomial potentials of the form , where
even, and . Mainly, we establish the limiting forms of global
spectral functions such as the zeta-regularized determinants and some spectral
zeta functions.Comment: latex text.tex, 3 files, 2 figures, 14 pages
http://www-spht.cea.fr/articles/T03/192 [SPhT-T03/192], submitted to Publ.
RIMS, Kyoto Univ. (special 40th anniversary issue
Zeta-regularisation for exact-WKB resolution of a general 1D Schr\"odinger equation
We review an exact analytical resolution method for general one-dimensional
(1D) quantal anharmonic oscillators: stationary Schr\"odinger 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.Comment: 17 pages, 2 figures. V2: minor amendments throughout, with one typo
corrected in eq.(19
A sharpening of Li's criterion for the Riemann Hypothesis
Exact and asymptotic formulae are displayed for the coefficients
used in Li's criterion for the Riemann Hypothesis. In particular, we argue that
if (and only if) the Hypothesis is true, for (with explicit and ). The approach also holds for more
general zeta or -functions.Comment: 1 Latex file, 5 pages, submitted to C.R. Acad. Sci. (Paris) S\'er. I.
V2: notation corrected in eq.(7); eq.(9) made more precis
Semi-classical correspondence and exact results : the case of the spectra of homogeneous Schrödinger operators [erratum: J. Physique Lett. 43 (1982) p.159]
Version française : Comptes Rendus de l'Académie des Sciences Série I, 293 (1981) pp.709-712The semi-classical WKB method is liable to yield exact results in full generality. Such results, in the case of the Schrödinger operator with a homogeneous potential , appear as a functional equation obeyed by the Fredholm determinant, or as arithmetical identities satisfied by the Zeta function of the spectrum
Zeta functions over zeros of Zeta functions and an exponential-asymptotic view of the Riemann Hypothesis
International audienceWe 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 repre-sentation for the Keiper–Li coefficients, whose large-order behavior thereby becomes computable by the method of steepest descents; then the dominant saddle-point en-tirely 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)
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.Comment: 16 p., 7 figs. Workshop for Prof. Yoshitsugu TAKEI's 60-th birthday,
RIMS, Kyoto Univ., Oct. 202
An exact solution method for 1D polynomial Schr\"odinger equations
Stationary 1D Schr\"odinger equations with polynomial potentials are reduced
to explicit countable closed systems of exact quantization conditions, which
are selfconsistent constraints upon the zeros of zeta-regularized spectral
determinants, complementing the usual asymptotic (Bohr--Sommerfeld)
constraints. (This reduction is currently completed under a certain vanishing
condition.) In particular, the symmetric quartic oscillators are admissible
systems, and the formalism is tested upon them. Enforcing the exact and
asymptotic constraints by suitable iterative schemes, we numerically observe
geometric convergence to the correct eigenvalues/functions in some test cases,
suggesting that the output of the reduction should define a contractive
fixed-point problem (at least in some vicinity of the pure case).Comment: flatex text.tex, 4 file
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