Simplifications of the Keiper/Li approach to the Riemann Hypothesis

The Keiper/Li constants $\{\lambda_n\}_{n=1,2,\ldots}$ 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.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 $(q^M + g q^N)$, 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.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 $\lambda_n$ used in Li's criterion for the Riemann Hypothesis. In particular, we argue that if (and only if) the Hypothesis is true, $\lambda_n \sim n(A \log n +B)$ for $n \to \infty$ (with explicit $A>0$ and $B$). The approach also holds for more general zeta or $L$-functions.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 $q^{2M}$, 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 $q^4$ case).Comment: flatex text.tex, 4 file