Related papers: The Riemann Zeta Function and Vacuum Spectrum

We have studied some properties of the special Gram points of the Riemann zeta function which lie on contour lines ${\bf Im}(\zeta ( s )) = 0$ which do not contain zeroes of $\zeta ( s )$. We find that certain functions of these points,…

We use the functorial properties of Rieffel's pseudodifferential calculus to study families of operators associated to topological dynamical systems acted by a symplectic space. Information about the spectra and the essential spectra are…

We report on some properties of the $\xi(s)$ function and its value on the critical line, $\Xi(t)=\xi\left(\tfrac{1}{2}+it\right)$. First, we present some identities that hold for the log derivatives of a holomorphic function. We then…

This brief note explicates some mathematical details of Phys. Rev. Lett. 118, 130201 (2017), by showing how a version of the operator of that paper can be rigorously constructed on a well-defined linear space of functions.

We construct in a Sonine Space of entire functions a subspace related to the Riemann zeta function and we show that the quotient contains vectors intrinsically attached to the non-trivial zeros and their multiplicities.

We found, by Hurwitz's Zeta Function, a new functional equation for Riemann Zeta Function. Considering this equation for $s=2$ and $s=1$, we determine a relation between the values of Riemann zeta Function on positive integers. The Matrix…

We study the value distribution of the Riemann zeta function near the line $\Re s = 1/2$. We find an asymptotic formula for the number of $a$-values in the rectangle $ 1/2 + h_1 / (\log T)^\theta \leq \Re s \leq 1/2+ h_2 /(\log T)^\theta $,…

We intimate deeper connections between the Riemann zeta and gamma functions than often reported and further derive a new formula for expressing the value of $\zeta(2n+1)$ in terms of zeta at other fractional points. This paper also…

W. Luo has investigated the distribution of zeros of the derivative of the Selberg zeta function associated to compact hyperbolic Riemann surfaces. In essence, the main results in Luo's article involve the following three points: Finiteness…

The paper develops a theory of spectral boundary value problems from the perspective of general theory of linear operators in Hilbert spaces. An abstract form of spectral boundary value problem with generalized boundary conditions is…

A system of functions (signals) on the finite line, called the oscillator system, is described and studied. Applications of this system for discrete radar and digital communication theory are explained. Keywords: Weil representation,…

In 1999 Berry and Keating showed that a regularization of the 1D classical Hamiltonian H = xp gives semiclassically the smooth counting function of the Riemann zeros. In this paper we first generalize this result by considering a phase…

The Zeeman-Hamilton operators of free charged particles are identified with the Laplacians of certain Riemannian manifolds, called Zeeman manifolds. The quantum Hilbert space decomposes into subspaces (Zeeman zones) which are invariant…

We introduce the notion of hidden quantum correlations. We present the mean values of observables depending on one classical random variable described by the probability distribution in the form of correlation functions of two (three, etc.)…

From eigensolutions of the harmonic oscillator or Kepler-Coulomb Hamiltonian we extend the functional equation for the Riemann zeta function and develop integral representations for the Riemann xi function that is the completed classical…

For the same potential as originally studied by Ma [Phys. Rev. {\bf 71}, 195 (1947)] we obtain analytic expressions for the Jost functions and the residui of the S-matrix of both (i) redundant poles and (ii) the poles corresponding to true…

In this paper we treat the classical Riemann zeta function as a function of three variables: one is the usual complex $\adyn$-dimensional, customly denoted as $s$, another two are complex infinite dimensional, we denote it as $\b =…

We prove an equivalent of the Riemann hypothesis in terms of the functional equation (in its asymmetrical form) and the $a$-points of the zeta-function, i.e., the roots of the equation $\zeta(s)=a$, where $a$ is an arbitrary fixed complex…

We continue to investigate the physical interpretation of the Riemann zeta function as a FZZT brane partition function associated with a matrix/gravity correspondence begun in arxiv:0708.0645. We derive the master matrix of the $(2,1)$…

We study the value-distribution of Dirichlet polynomials on the critical line $\Re(s)=\tfrac{1}{2}$. As a consequence, we prove a corollary on small consecutive gaps between zeros of the Riemann zeta function. We also examine the…