Related papers: The Riemann Zeta Function and Vacuum Spectrum
We represent the Riemann zeta function in the half-plane $\Re s >1$ via series whose terms admit geometrically decreasing bounds. Due to an underlying recurrence relation, which is used to compute coefficients entering into the terms, the…
For an operator of a certain class in Hilbert space, we introduce axioms of an abstract intersection theory, which we prove to be equivalent to the Riemann Hypothesis concerning the spectrum of that operator. In particular if the nontrivial…
Some computations made about the Riemann Hypothesis and in particular, the verification that zeroes of zeta belong on the critical line and the extension of zero-free region are useful to get better effective estimates of number theory…
The random matrix ensembles are applied to the quantum statistical two-dimensional systems of electrons. The quantum systems are studied using the finite dimensional real, complex and quaternion Hilbert spaces of the eigenfunctions. The…
For an arbitrary complex number $a\neq 0$ we consider the distribution of values of the Riemann zeta-function $\zeta$ at the $a$-points of the function $\Delta$ which appears in the functional equation $\zeta(s)=\Delta(s)\zeta(1-s)$. These…
Berry and Keating conjectured that the classical Hamiltonian H = xp is related to the Riemann zeros. A regularization of this model yields semiclassical energies that behave, in average, as the non trivial zeros of the Riemann zeta…
The Riemann Hypothesis can be reformulated as statements about the eigenvalues of certain matrices whose entries are defined in terms of the Taylor coefficients of the zeta function. These eigenvalues exhibit interesting visual patterns…
We extend the application of the techniques developed within the framework of the pseudo-Hermitian quantum mechanics to study a unitary quantum system described by an imaginary PT-symmetric potential v(x) having a continuous real spectrum.…
In this article, we provide a unified framework for studying the convergence of rescaled characteristic polynomials of random matrices from various classical ensembles as well as functional convergence results for the Riemann zeta function.…
The Riemann theta function is a complex-valued function of g complex variables. It appears in the construction of many (quasi-) periodic solutions of various equations of mathematical physics. In this paper, algorithms for its computation…
I present two independent proofs of the Riemann Hypothesis considered by many the greatest unsolved problem in mathematics. I find that the admissible domain of complex zeros of the Riemann Zeta Function is the critical line. The methods…
We investigate the random dynamics of rational maps on the Riemann sphere and the dynamics of semigroups of rational maps on the Riemann sphere. We show that regarding random complex dynamics of polynomials, in most cases, the chaos of the…
In this paper we consider two spectral realizations of the zeros of the Riemann zeta function. The first one involves all non-trivial (non-real) zeros and is expressed in terms of a Laplacian intimately related to the prolate wave operator.…
The individual terms of the series representing the Riemann zeta function are examined geometrically from their accumulated plot in the complex plane. Symmetry is identified and determined mathematically for comparison with more traditional…
In this paper,using methods of weight functions and techniques of real analysis, we provide a multidimensional Hilbert-type integral inequality with a homogeneous kernel of degree 0 as well as a best possible constant factor related to the…
Let $\gamma$ denote imaginary parts of complex zeros of the Riemann zeta-function $\zeta(s)$. Certain sums over the $\gamma$'s are evaluated, by using the function $G(s) = \sum_{\gamma>0}\gamma^{-s}$ and other techniques. Some integrals…
Global mapping properties of the Riemann Zeta function are used to investigate its non trivial zeros.
In this short letter, we reformulate the Riemann zeta function using the holographic framework of the celestial conformal field theory. For spacetime dimension larger than our Minkowski spacetime $M^4$, the Riemann zeta function is…
This paper compares the distribution of zeros of the Riemann zeta function $\zeta(s)$ with those of a symmetric combination of zeta functions, denoted ${\cal T}_+(s)$, known to have all its zeros located on the critical line $\Re(s)=1/2$.…
A new integral representation for the Riemann zeta function is derived. This representation covers the important region of the complex plane where the real part of the argument of the function lies between 0 and 1. Using this…