scholarly journals The Riemann Zeros as Spectrum and the Riemann Hypothesis

Symmetry ◽  
2019 ◽  
Vol 11 (4) ◽  
pp. 494 ◽  
Author(s):  
Germán Sierra
Keyword(s):  
Zeta Function ◽  
Experimental Observation ◽  
Riemann Hypothesis ◽  
Bound States ◽  
Critical Line ◽  
Delta Function ◽  
Dirac Fermion ◽  
Adjoint Extension ◽  
Massless Dirac Fermion

We present a spectral realization of the Riemann zeros based on the propagation of a massless Dirac fermion in a region of Rindler spacetime and under the action of delta function potentials localized on the square free integers. The corresponding Hamiltonian admits a self-adjoint extension that is tuned to the phase of the zeta function, on the critical line, in order to obtain the Riemann zeros as bound states. The model suggests a proof of the Riemann hypothesis in the limit where the potentials vanish. Finally, we propose an interferometer that may yield an experimental observation of the Riemann zeros.

scholarly journals The Values of the Periodic Zeta-Function at the Nontrivial Zeros of Riemann’s Zeta-Function

Symmetry ◽  
2021 ◽  
Vol 13 (12) ◽  
pp. 2410
Author(s):  
Janyarak Tongsomporn ◽  
Saeree Wananiyakul ◽  
Jörn Steuding
Keyword(s):  
Asymptotic Formula ◽  
Zeta Function ◽  
Riemann Zeta Function ◽  
Real Line ◽  
Riemann Hypothesis ◽  
Critical Line ◽  
Nontrivial Zeros ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
Riemann’S Zeta Function

In this paper, we prove an asymptotic formula for the sum of the values of the periodic zeta-function at the nontrivial zeros of the Riemann zeta-function (up to some height) which are symmetrical on the real line and the critical line. This is an extension of the previous results due to Garunkštis, Kalpokas, and, more recently, Sowa. Whereas Sowa’s approach was assuming the yet unproved Riemann hypothesis, our result holds unconditionally.

scholarly journals Experimental observation of two massless Dirac-fermion gases in graphene-topological insulator heterostructure

2D Materials ◽  
2016 ◽  
Vol 3 (2) ◽  
pp. 021009 ◽  
Author(s):  
Guang Bian ◽  
Ting-Fung Chung ◽  
Chaoyu Chen ◽  
Chang Liu ◽  
Tay-Rong Chang ◽  
...  
Keyword(s):  
Experimental Observation ◽  
Topological Insulator ◽  
Dirac Fermion ◽  
Massless Dirac Fermion

scholarly journals Some Remarks and Propositions on Riemann Hypothesis

2020 ◽  
Author(s):  
Jamal Salah
Keyword(s):  
Functional Equation ◽  
Integral Representation ◽  
Zeta Function ◽  
Riemann Zeta Function ◽  
Riemann Hypothesis ◽  
Critical Line ◽  
Equal Zero ◽  
Riemann Zeta

In this article we look at some well know results of Riemann Zeta function in a different light. We explore the proofs of Zeta integral Representation, Analytic continuity and the first functional equation. We propound some modifications in order to reasonably justify the location of the non-trivial zeros on the critical line: s= 1/2 by assuming that ζ(s) and ζ(1-s) simultaneously equal zero

scholarly journals Riemann Hypothesis and Random Walks: The Zeta Case

Symmetry ◽  
2021 ◽  
Vol 13 (11) ◽  
pp. 2014
Author(s):  
André LeClair
Keyword(s):  
Zeta Function ◽  
Riemann Hypothesis ◽  
Critical Line ◽  
Dirichlet Characters ◽  
The Riemann Zeta Function ◽  
Point Correlation ◽  
Point Correlation Function ◽  
Riemann Zeta ◽  
The Right ◽  
Very High

In previous work, it was shown that if certain series based on sums over primes of non-principal Dirichlet characters have a conjectured random walk behavior, then the Euler product formula for its L-function is valid to the right of the critical line ℜ(s)>12, and the Riemann hypothesis for this class of L-functions follows. Building on this work, here we propose how to extend this line of reasoning to the Riemann zeta function and other principal Dirichlet L-functions. We apply these results to the study of the argument of the zeta function. In another application, we define and study a one-point correlation function of the Riemann zeros, which leads to the construction of a probabilistic model for them. Based on these results we describe a new algorithm for computing very high Riemann zeros, and we calculate the googol-th zero, namely 10100-th zero to over 100 digits, far beyond what is currently known. Of course, use is made of the symmetry of the zeta function about the critical line.

scholarly journals Riemann Hypothesis Holds virtually true

2021 ◽  
Author(s):  
Jamal Salah
Keyword(s):  
Zeta Function ◽  
Riemann Hypothesis ◽  
Critical Line ◽  
Simple Pole ◽  
Equal Zero

If infinity times zero equal zero then Zeta function has an analytic continuity over the whole complex plan except a simple pole at 1. Since infinity times zero is undefined, then Riemann' s approach remain not sharp. However, it is true that the non trivial zeros lie at the critical line x = 1/2 despite there simultaneous virtual existence or in another words, such zeros are assumed to exist.

scholarly journals Fourier coefficients associated with the Riemann zeta-function

2016 ◽  
Vol 8 (1) ◽  
pp. 16-20
Author(s):  
Yu.V. Basiuk ◽  
S.I. Tarasyuk
Keyword(s):  
Fourier Series ◽  
Zeta Function ◽  
Riemann Zeta Function ◽  
Riemann Hypothesis ◽  
Fourier Coefficients ◽  
Critical Line ◽  
Series Method ◽  
Fourier Series Method ◽  
The Riemann Zeta Function ◽  
Riemann Zeta

We study the Riemann zeta-function $\zeta(s)$ by a Fourier series method. The summation of $\log|\zeta(s)|$ with the kernel $1/|s|^{6}$ on the critical line $\mathrm{Re}\; s = \frac{1}{2}$ is the main result of our investigation. Also we obtain a new restatement of the Riemann Hypothesis.

On the Size of an Expression in the Nyman–Beurling-Báez–Duarte Criterion for the Riemann Hypothesis

2018 ◽  
Vol 61 (3) ◽  
pp. 622-627
Author(s):  
Helmut Maier ◽  
Michael Th. Rassias
Keyword(s):  
Zeta Function ◽  
Riemann Zeta Function ◽  
Crucial Role ◽  
Riemann Hypothesis ◽  
Critical Line ◽  
Nontrivial Zeros ◽  
The Riemann Zeta Function ◽  
Dirichlet Polynomials ◽  
Riemann Zeta

AbstractA crucial role in the Nyman–Beurling–Báez-Duarte approach to the Riemann Hypothesis is played by the distancewhere the infimum is over all Dirichlet polynomialsof length N. In this paper we investigate under the assumption that the Riemann zeta function has four nontrivial zeros off the critical line.

scholarly journals On the value-distribution of iterated integrals of the logarithm of the Riemann zeta-function I: Denseness

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Kenta Endo ◽  
Shōta Inoue
Keyword(s):  
Zeta Function ◽  
Complex Plane ◽  
Riemann Zeta Function ◽  
Open Problem ◽  
Riemann Hypothesis ◽  
Critical Line ◽  
Iterated Integrals ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
Iterated Integral

AbstractWe consider iterated integrals of {\log\zeta(s)} on certain vertical and horizontal lines. Here, the function {\zeta(s)} is the Riemann zeta-function. It is a well-known open problem whether or not the values of the Riemann zeta-function on the critical line are dense in the complex plane. In this paper, we give a result for the denseness of the values of the iterated integrals on the horizontal lines. By using this result, we obtain the denseness of the values of {\int_{0}^{t}\log\zeta(\frac{1}{2}+it^{\prime})\,dt^{\prime}} under the Riemann Hypothesis. Moreover, we show that, for any {m\geq 2}, the denseness of the values of an m-times iterated integral on the critical line is equivalent to the Riemann Hypothesis.

scholarly journals An Analytical and Numerical Detour for the Riemann Hypothesis

Information ◽  
2021 ◽  
Vol 12 (11) ◽  
pp. 483
Author(s):  
Michel Riguidel
Keyword(s):  
Riemann Hypothesis ◽  
Meromorphic Functions ◽  
Critical Line ◽  
Numerical Approach ◽  
Product Formula ◽  
Critical Strip ◽  
Associated Functions ◽  
Poles And Zeros ◽  
Riemann’S Zeta Function ◽  
Insight Into

From the functional equation of Riemann’s zeta function, this article gives new insight into Hadamard’s product formula. The function and its family of associated functions, expressed as a sum of rational fractions, are interpreted as meromorphic functions whose poles are the poles and zeros of the function. This family is a mathematical and numerical tool which makes it possible to estimate the value of the function at a point in the critical strip from a point on the critical line .Generating estimates of at a given point requires a large number of adjacent zeros, due to the slow convergence of the series. The process allows a numerical approach of the Riemann hypothesis (RH). The method can be extended to other meromorphic functions, in the neighborhood of isolated zeros, inspired by the Weierstraß canonical form. A final and brief comparison is made with the and functions over finite fields.

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