Quantum mechanical potentials related to the prime numbers and Riemann zeros

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dc.creatorSchumayer, Daniel
dc.creatorvan Zyl, Brandon P.
dc.creatorHutchinson, David A. W.
dc.date2008-11-10
dc.date.accessioned2026-07-07T12:05:11Z
dc.date.available2026-07-07T12:05:11Z
dc.descriptionPrime numbers are the building blocks of our arithmetic, however, their distribution still poses fundamental questions. Bernhard Riemann showed that the distribution of primes could be given explicitly if one knew the distribution of the non-trivial zeros of the Riemann $ζ(s)$ function. According to the Hilbert-P{ó}lya conjecture there exists a Hermitean operator of which the eigenvalues coincide with the real part of the non-trivial zeros of $ζ(s)$. This idea encourages physicists to examine the properties of such possible operators, and they have found interesting connections between the distribution of zeros and the distribution of energy eigenvalues of quantum systems. We apply the Mar{č}henko approach to construct potentials with energy eigenvalues equal to the prime numbers and to the zeros of the $ζ(s)$ function. We demonstrate the multifractal nature of these potentials by measuring the R{é}nyi dimension of their graphs. Our results offer hope for further analytical progress.
dc.description7 pages, 5 figures, 2 tables
dc.identifierhttps://arxiv.org/abs/0811.1389
dc.identifierhttp://arxiv.org/abs/0811.1389
dc.identifierPhysical Review E, vol. 78(5), 056215 (2008)
dc.identifierdoi:10.1103/PhysRevE.78.056215
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/208310
dc.subjectMathematical Physics
dc.titleQuantum mechanical potentials related to the prime numbers and Riemann zeros
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