The Selberg Trace Formula for PSL(2,ℝ) Volume I

Hejhal, Dennis A.

doi:10.1007/bfb0079608

Cited by 243 publications

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“…13) Similarly to (2 . 2) it is defined as the product but not over primes but over all primitive periodic orbits (ppo) for the motion on the surface considered…”

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confidence: 99%

Riemann Zeta Function and Quantum Chaos

Bogomolny

2007

Prog. Theor. Phys. Suppl.

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A brief review of recent developments in the theory of the Riemann zeta function inspired by ideas and methods of quantum chaos is given. §1. IntroductionAt the first glance the Riemann zeta function and quantum chaos are completely disjoint fields. The Riemann zeta function is a part of pure number theory but quantum chaos is a branch of theoretical physics devoted to the investigation of non-integrable quantum problems like the hydrogen atom in external fields.Nevertheless for a long time it was understood that there exist multiple interrelations between these two subjects. 1), 2) In § §2 and 3 the Riemann and the Selberg zeta functions and their trace formulae are informally compared. 3) From the comparison it appears that in many aspects zeros of the Riemann zeta function resemble eigenvalues of an unknown quantum chaotic Hamiltonian.One of the principal tools in quantum chaos is the investigation of statistical properties of deterministic energy levels of a given Hamiltonian. In such approach one stresses not precise values of physical quantities but their statistics by considering them as different realizations of a statistical ensemble. According to the BGS conjecture 4) energy levels of chaotic quantum systems have the same statistical properties as eigenvalues of standard random matrix ensembles depended only on the exact symmetries. In §4 it is argued that it is quite natural to conjecture that statistical properties of the Riemann zeros are the same as of eigenvalues of the gaussian unitary ensemble of random matrices (GUE). This conjecture is very well confirmed by numerics but only partial rigorous results are available. 5) In §5 a semiclassical method which permits, in principle, to calculate correlation functions is shortly discussed. The main problem here is to control correlations between periodic orbits with almost the same lengths. In § §6 and 7 it is demonstrated how the Hardy-Littlewood conjecture 6) about distribution of near-by primes leads to explicit formula for the two-point correlation function of the Riemann zeros. 7) The resulting formula describes non-universal approach to the GUE result in excellent agreement with numerical results.In §8 it is demonstrated how to calculate non-universal corrections to the nearestneighbor distribution for the Riemann zeros. 8) Spectral statistics is not the only interesting statistical characteristics of zeta functions. The mean moments of the Riemann zeta function along the critical line is another important subject that attracts wide attention in number theory during a by guest on April 14, 2015 http://ptps.oxfordjournals.org/ Downloaded from by guest on April 14, 2015 http://ptps.oxfordjournals.org/ Downloaded from

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“…13) Similarly to (2 . 2) it is defined as the product but not over primes but over all primitive periodic orbits (ppo) for the motion on the surface considered…”

mentioning

confidence: 99%

Riemann Zeta Function and Quantum Chaos

Bogomolny

2007

Prog. Theor. Phys. Suppl.

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“…It turns out that Selberg's zeta function is closely related to the eigenvalues of the Laplace operator on M (cf. Theorem II.4.10 and II.4.11 in [19]). (b) Let β be a real number ≥ 2.…”

Section: Selberg's Zeta Function and Trace Formulamentioning

confidence: 91%

Statistical properties of zeta functions’ zeros

Kargin¹

2014

Probab. Surveys

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“…The 2 and gives a spectrum with spikes at ordinates equal to the imaginary part of the Riemann zeta function zeros (Hejhal 1976).…”

Section: Imentioning

confidence: 99%

Detection and Discrimination of the Periodicity of Prime Numbers by Discrete Fourier Transform – Symphony of Primes

Csóka

2015

J. Adv. App. Comput. Math.

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A novel representation of a quasi-periodic modified von Mangoldt function ( ) on prime numbers and its decomposition into Fourier series has been investigated. We focus on some particular quantities characterizing the modified von Mangoldt function. The results indicate that prime number progression can be decomposed into periodic sequences. The main approach is to decompose it into sin or cosine function. Basically, it is applied to extract hidden periodicities in seemingly quasi periodic prime function. Numerical evidences were provided to confirm the periodic distribution of primes.

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The Selberg Trace Formula for PSL(2,ℝ) Volume I

Cited by 243 publications

References 0 publications

Riemann Zeta Function and Quantum Chaos

Riemann Zeta Function and Quantum Chaos

Statistical properties of zeta functions’ zeros

Detection and Discrimination of the Periodicity of Prime Numbers by Discrete Fourier Transform – Symphony of Primes

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