The argument of the Riemann zeta function

Карацуба, Анатолий Алексеевич; Korolev, Maxim Aleksandrovich

doi:10.1070/rm2005v060n03abeh000848

Cited by 45 publications

(64 citation statements)

References 28 publications

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“…In their semiclassical analysis, Wu and Sprung [1] took into account only the leading term of the Riemann-von Mangoldt formula [13], but nevertheless, as our results indicate, doing so is able to yield, via the Schrödinger equation, eigenvalues that quite closely approximate the Riemann zeros themselves. It might prove beneficial, in trying to account for the (relatively small) residual variation -especially since we have been working in a nonasymptotic regime -to extend the Wu-Sprung approach by incorporating the higher order nonoscillatory terms of the Riemann-von Mangoldt formula too (cf.…”

Section: Higher Order Corrections To the Wu-sprung Potentialmentioning

confidence: 92%

“…17.) The remaining part (having both smooth and nonsmooth components) not utilized by Wu and Sprung is expressible as [13] S(E)…”

Section: Higher Order Corrections To the Wu-sprung Potentialmentioning

confidence: 99%

“…In Sect. 3, we demonstrate how to incorporate more terms of the Riemann-von Mangoldt formula [13] for the cumulated number of Riemann zeros than Wu-Sprung themselves did, using a semiclassical argument, in deriving x WS (V ). (It remains, however, to implement numerically these last findings.…”

Section: Introductionmentioning

confidence: 99%

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Fractal fits to Riemann zeros

Slater¹

2007

Can. J. Phys.

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Wu and Sprung (Phys. Rev. E, 48, 2595 (1993)) reproduced the first 500 nontrivial Riemann zeros, using a one-dimensional local potential model. They concluded -as did van Zyl and Hutchinson (Phys. Rev. E, 67, 066211 (2003)) -that the potential possesses a fractal structure of dimension d = 3/2. We model the nonsmooth fluctuating part of the potential by the alternating-sign sine series fractal of Berry and Lewis A(x, γ ). Setting d = 3/2, we estimate the frequency parameter (γ ), plus an overall scaling parameter (σ ) that we introduce. We search for that pair of parameters (γ, σ ) that minimizes the least-squares fit S n (γ , σ ) of the lowest n eigenvalues -obtained by solving the one-dimensional stationary (nonfractal) Schrödinger equation with the trial potential (smooth plus nonsmooth parts) -to the lowest n Riemann zeros for n = 25. For the additional cases, we study, n = 50 and 75, we simply set σ = 1. The fits obtained are compared to those found by using just the smooth part of the Wu-Sprung potential without any fractal supplementation. Some limited improvement -5.7261 versus 6.392 07 (n = 25), 11.2672 versus 11.7002 (n = 50), and 16.3119 versus 16.6809 (n = 75) -is found in our (nonoptimized, computationally bound) search procedures. The improvements are relatively strong in the vicinities of γ = 3 and (its square) 9. Further, we extend the Wu-Sprung semiclassical framework to include higher order corrections from the Riemann-von Mangoldt formula (beyond the leading, dominant term) into the smooth potential. (2003)) -que le potentiel possède une structure fractale de dimension d = 3/2. Nous modélisons la partie de ce potentiel qui n'est pas douce à l'aide de la série sine fractale à signe alterné de Berry et Lewis A(x, y). Fixant d = 3/2, nous estimons le paramètre de fréquence (γ ), en plus d'un paramètre global d'échelle (σ ) que nous introduisons. Nous recherchons la paire de paramètres (γ , σ ) qui minimise l'ajustement en moindres carrés S n (γ , σ ) des n plus basses valeurs propresobtenues en solutionnant l'équation de Schrodinger stationnaire non fractale à une dimension avec le potentiel d'essai (parties douce et non douce) -aux n plus bas zéros de Riemann pour n = 25. Nous étudions aussi les cas n = 50 et 75, avec σ = 1. Les résultats obtenus sont comparés à ceux obtenus en utilisant seulement la partie douce du potentiel de WuSprung sans aucun ajout fractal. Nous observons une certaine amélioration -5,7261 versus 6,392 07 (n = 25), 11,2672 versus 11,7002 (n = 50) et 16,3119 versus 16,6809 (n = 75) (dans notre recherche qui n'est pas optimisée et a des limites de calcul). Les améliorations sont relativement importantes au voisinage de γ = 3 et 9 (son carré). De plus, nous avons étendu le cadre de l'approche semi-classique de Wu-Sprung pour inclure des corrections PACS

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Section: Higher Order Corrections To the Wu-sprung Potentialmentioning

confidence: 92%

“…17.) The remaining part (having both smooth and nonsmooth components) not utilized by Wu and Sprung is expressible as [13] S(E)…”

Section: Higher Order Corrections To the Wu-sprung Potentialmentioning

confidence: 99%

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Fractal fits to Riemann zeros

Slater¹

2007

Can. J. Phys.

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“…Also, the strongly oscillatory behavior of S(E) forces the potential V (x) to be a multi-valued function of x. The expression for ∆(E) is [33] ∆(E) = E 4 log 1 + 1 The graph of the N (E) level counting function is displayed in Fig. 1.…”

Section: Riemann Hypothesis and Bohr-sommerfeld Quantizationmentioning

confidence: 99%

“…An extensive analysis of the behavior of S(E) can be found in [33]. In particular the property S(E) is a piecewise smooth function with discontinuities at the ordinates E n of the complex zeroes of ζ(s n = 1 2 + iE n ) = 0.…”

Section: Riemann Hypothesis and Bohr-sommerfeld Quantizationmentioning

confidence: 99%

The search for a Hamiltonian whose energy spectrum coincides with the Riemann zeta zeroes

Aschheim

Perelman

Irwin

2017

Int. J. Geom. Methods Mod. Phys.

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Inspired by the Hilbert-Polya proposal to prove the Riemann Hypothesis we have studied the Schroedinger QM equation involving a highly nontrivial potential, and whose self-adjoint Hamiltonian operator has for its energy spectrum one which approaches the imaginary parts of the zeta zeroes only in the asymptotic (very large N ) region. The ordinates λn are the positive imaginary parts of the nontrivial zeta zeroes in the critical line : sn = 1 2 + iλn. The latter results are consistent with the validity of the Bohr-Sommerfeld semi-classical quantization condition. It is shown how one may modify the parameters which define the potential, and fine tune its values, such that the energy spectrum of the (modified) Hamiltonian matches not only the first two zeroes but the other consecutive zeroes. The highly nontrivial functional form of the potential is found via the Bohr-Sommerfeld quantization formula using the full-fledged Riemann-von Mangoldt counting formula (without any truncations) for the number N (E) of zeroes in the critical strip with imaginary part greater than 0 and less than or equal to E.

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Riemann zeros as quantized energies of scattering with impurities

LeClair,

Mussardo

2024

J. High Energ. Phys.

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We construct an integrable physical model of a single particle scattering with impurities spread on a circle. The S-matrices of the scattering with the impurities are such that the quantized energies of this system, coming from the Bethe Ansatz equations, correspond to the imaginary parts of the non-trivial zeros of the the Riemann ζ(s) function along the axis $$\mathfrak{R}\left(s\right)=\frac{1}{2}$$ of the complex s-plane. A simple and natural generalization of the original scattering problem leads instead to Bethe Ansatz equations whose solutions are the non-trivial zeros of the Dirichlet L-functions again along the axis $$\mathfrak{R}\left(s\right)=\frac{1}{2}$$. The conjecture that all the non-trivial zeros of these functions are aligned along this axis of the complex s-plane is known as the Generalised Riemann Hypothesis (GRH). In the language of the scattering problem analysed in this paper the validity of the GRH is equivalent to the completeness of the Bethe Ansatz equations. Moreover the idea that the validity of the GRH requires both the duality equation (i.e. the mapping s → 1 – s) and the Euler product representation of the Dirichlet L-functions finds additional and novel support from the physical scattering model analysed in this paper. This is further illustrated by an explicit counterexample provided by the solutions of the Bethe Ansatz equations which employ the Davenport-Heilbronn function $$\mathcal{D}\left(s\right)$$, i.e. a function whose completion satisfies the duality equation χ(s) = χ(1 – s) but that does not have an Euler product representation. In this case, even though there are infinitely many solutions of the Bethe Ansatz equations along the axis $$\mathfrak{R}\left(s\right)=\frac{1}{2}$$, there are also infinitely many pairs of solutions away from this axis and symmetrically placed with respect to it.

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The argument of the Riemann zeta function

Cited by 45 publications

References 28 publications

Fractal fits to Riemann zeros

Fractal fits to Riemann zeros

The search for a Hamiltonian whose energy spectrum coincides with the Riemann zeta zeroes

Riemann zeros as quantized energies of scattering with impurities

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