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Prime 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 $zeta(s)$ function. According to the Hilbert-P{o}lya conjecture there exists a Hermitean operator of which the eigenvalues coincide with the real part of the non-trivial zeros of $zeta(s)$. 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{v{c}}henko approach to construct potentials with energy eigenvalues equal to the prime numbers and to the zeros of the $zeta(s)$ function. We demonstrate the multifractal nature of these potentials by measuring the R{e}nyi dimension of their graphs. Our results offer hope for further analytical progress.
The Riemann hypothesis states that all nontrivial zeros of the zeta function lie in the critical line $Re(s)=1/2$. Hilbert and Polya suggested that one possible way to prove the Riemann hypothesis is to interpret the nontrivial zeros in the light of
The spherically symmetric potential $a ,delta (r-r_0)+b,delta (r-r_0)$ is generalised for the $d$-dimensional space as a characterisation of a unique selfadjoint extension of the free Hamiltonian. For this extension of the Dirac delta, the spectrum
We present a remarkable connection between the asymptotic behavior of the Riemann zeros and one-loop effective action in Euclidean scalar field theory. We show that in a two-dimensional space, the asymptotic behavior of the Fourier transform of two-p
We prove quantum ergodicity for a family of graphs that are obtained from ergodic one-dimensional maps of an interval using a procedure introduced by Pakonski et al (J. Phys. A, v. 34, 9303-9317 (2001)). As observables we take the L^2 functions on th
An eigenvalue problem relevant for non-linear sigma model with singular metric is considered. We prove the existence of a non-degenerate pure point spectrum for all finite values of the size R of the system. In the infrared (IR) regime (large R) the