scholarly journals Majorana quanta, string scattering, curved spacetimes and the Riemann Hypothesis

2021 ◽  
Author(s):  
Fabrizio Tamburini ◽  
Ignazio Licata
Keyword(s):  
Riemann Hypothesis ◽  
Open String ◽  
Prime Numbers ◽  
Dirac Particles ◽  
Ideal System ◽  
S Matrix ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
Poles And Zeros ◽  
The Relationship

Abstract The Riemann Hypothesis states that the Riemann zeta function ζ(z) admits a set of “non-trivial” zeros that are complex numbers supposed to have real part 1/2. Their distribution on the complex plane is thought to be the key to determine the number of prime numbers before a given number. Hilbert and Pólya suggested that the Riemann Hypothesis could be solved through the mathematical tools of physics, finding a suitable Hermitian or unitary operator that describe classical or quantum systems, whose eigenvalues distribute like the zeros of ζ(z). A different approach is that of finding a correspondence between the distribution of the ζ(z) zeros and the poles of the scattering matrix S of a physical system. Our contribution is articulated in two parts: in the first we apply the infinite-components Majorana equation in a Rindler spacetime and compare the results with those obtained with a Dirac particle following the Hilbert-Pólya approach showing that the Majorana solution has a behavior similar to that of massless Dirac particles and finding a relationship between the zeros of zeta end the energy states. Then, we focus on the S-matrix approach describing the bosonic open string scattering for tachyonic states with the Majorana equation. Here we find that, thanks to the relationship between the angular momentum and energy/mass eigenvalues of the Majorana solution, one can explain the still unclear point for which the poles and zeros of the S-matrix of an ideal system that can satisfy the Riemann Hypothesis, exist always in pairs and are related via complex conjugation. As claimed in the literature, if this occurs and the claim is correct, then the Riemann Hypothesis could be in principle satisfied, tracing a route to a proof.

A Proof to the Riemann Hypothesis

2020 ◽  
Author(s):  
Sourangshu Ghosh
Keyword(s):  
Number Theory ◽  
Zeta Function ◽  
Riemann Zeta Function ◽  
Riemann Hypothesis ◽  
Prime Numbers ◽  
Complex Numbers ◽  
The Riemann Zeta Function ◽  
Riemann Zeta

In this paper, we shall try to prove the Riemann Hypothesis which is a conjecture that the Riemann zeta function hasits zeros only at the negative even integers and complex numbers with real part ½. This conjecture is very importantand of considerable interest in number theory because it tells us about the distribution of prime numbers along thereal line. This problem is one of the clay mathematics institute’s millennium problems and also comprises the 8ththe problem of Hilbert’s famous list of 23 unsolved problems. There have been many unsuccessful attempts in provingthe hypothesis. In this paper, we shall give proof to the Riemann Hypothesis.

scholarly journals Two New Equivalent Statements of Riemann Hypothesis

2020 ◽  
Author(s):  
Jamal Salah
Keyword(s):  
Zeta Function ◽  
Riemann Hypothesis ◽  
Univalent Functions ◽  
Prime Numbers ◽  
Regular Pattern ◽  
Complete Proof ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
Interesting Solution ◽  
Shed Light

The distribution of such prime numbers among all natural numbersdoes not follow any regular pattern; however, the Germanmathematician G. F. B. Riemann (1826-1866) observed that thefrequency of prime numbers is very closely related to the behaviorof an elaborate function called the Riemann zeta function s. Thenontrivial zeroes of zeta function have 1 2 as their real part. This hasbeen checked for the first 1,500,000,000 solutions. A proof that it istrue for every interesting solution would shed light on many ofthe mysteries surrounding the distribution of prime numbers.The celebrated Riemann hypothesis remains unsolved since it wasintroduced in 1859. Miscellaneous approaches have been consideredwithout any exact and complete proof. Furthermore, some equivalentstatements have been established.In this work, we consider the famous Robin inequality and propound aconnection to the theory of univalent functions by the means of Koebefunction.

ON THE EXPLICIT FORMULA IN THE THEORY OF PRIME NUMBERS

2012 ◽  
Vol 08 (03) ◽  
pp. 589-597 ◽  
Author(s):  
XIAN-JIN LI
Keyword(s):  
New York ◽  
Zeta Function ◽  
Explicit Formula ◽  
Riemann Zeta Function ◽  
Riemann Hypothesis ◽  
Prime Numbers ◽  
Nontrivial Zeros ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
Li's Criterion

In [Complements to Li's criterion for the Riemann hypothesis, J. Number Theory77 (1999) 274–287] Bombieri and Lagarias observed the remarkable identity [1 - (1 - 1/s)n] + [1 - (1 - 1/(1 - s))n] = [1 - (1 - 1/s)n]⋅[1 - (1 - 1/(1 - s))n], and pointed out that the positivity in Li's criterion [The positivity of a sequence of numbers and the Riemann hypothesis, J. Number Theory65 (1997) 325–333] has the same meaning as in Weil's criterion [Sur les "formules explicites" de la théorie des nombres premiers, in Oeuvres Scientifiques, Collected Paper, Vol. II (Springer-Verlag, New York, 1979), pp. 48–61]. Let λn = ∑ρ[1 - (1 - 1/ρ)n] for n = 1, 2, …, where ρ runs over the complex zeros of the Riemann zeta function ζ(s). In this note, a certain truncation of λn is expressed as Weil's explicit formula [Sur les "formules explicites" de la théorie des nombres premiers, in Oeuvres Scientifiques, Collected Paper, Vol. II (Springer-Verlag, New York, 1979), pp. 48–61] for each positive integer n. By using the Bombieri and Lagarias' identity, we prove that the positivity of these truncations implies the Riemann hypothesis. If these truncations have suitable upper bounds, we prove that all nontrivial zeros of the Riemann zeta function lie on the critical line.

scholarly journals A Schrödinger Equation for Solving the Riemann Hypothesis

2017 ◽  
Author(s):  
Frederick Ira Moxley III
Keyword(s):  
Zeta Function ◽  
Analytic Continuation ◽  
Riemann Zeta Function ◽  
Riemann Hypothesis ◽  
Prime Numbers ◽  
Quantum Systems ◽  
Dinger Equation ◽  
Nontrivial Zeros ◽  
The Riemann Zeta Function ◽  
Riemann Zeta

The Hamiltonian of a quantum mechanical system has an affiliated spectrum. If this spectrum is the sequence of prime numbers, a connection between quantum mechanics and the nontrivial zeros of the Riemann zeta function can be made. In this case, the Riemann zeta function is analogous to chaotic quantum systems, as the harmonic oscillator is for integrable quantum systems. Such quantum Riemann zeta function analogies have led to the Bender-Brody-Müller (BBM) conjecture, which involves a non-Hermitian Hamiltonian that maps to the zeros of the Riemann zeta function. If the BBM Hamiltonian can be shown to be Hermitian, then the Riemann Hypothesis follows. As such, herein we perform a symmetrization procedure of the BBM Hamiltonian to obtain a unique Hermitian Hamiltonian that maps to the zeros of the analytic continuation of the Riemann zeta function, and provide an analytical expression for the eigenvalues of the results. The holomorphicity of the resulting eigenvalues is examined. Moreover, a second quantization of the resulting Schrödinger equation is performed, and a convergent solution for the nontrivial zeros of the analytic continuation of the Riemann zeta function is obtained. Finally, from the holomorphicity of the eigenvalues it is shown that the real part of every nontrivial zero of the Riemann zeta function converges at σ = 1/2.

scholarly journals A Schrödinger Equation for Solving the Riemann Hypothesis

2017 ◽  
Author(s):  
Frederick Ira Moxley
Keyword(s):  
Zeta Function ◽  
Analytic Continuation ◽  
Riemann Zeta Function ◽  
Riemann Hypothesis ◽  
Prime Numbers ◽  
Quantum Systems ◽  
Dinger Equation ◽  
Nontrivial Zeros ◽  
The Riemann Zeta Function ◽  
Riemann Zeta

The Hamiltonian of a quantum mechanical system has an affiliated spectrum. If this spectrum is the sequence of prime numbers, a connection between quantum mechanics and the nontrivial zeros of the Riemann zeta function can be made. In this case, the Riemann zeta function is analogous to chaotic quantum systems, as the harmonic oscillator is for integrable quantum systems. Such quantum Riemann zeta function analogies have led to the Bender-Brody-Mu¨ller (BBM) conjecture, which involves a non-Hermitian Hamiltonian that maps to the zeros of the Riemann zeta function. If the BBM Hamiltonian can be shown to be Hermitian, then the Riemann Hypothesis follows. As such, herein we perform a symmetrization procedure of the BBM Hamiltonian to obtain a unique Hermitian Hamiltonian that maps to the zeros of the analytic continuation of the Riemann zeta func- tion, and provide an analytical expression for the eigenvalues of the results. The holomorphicity of the resulting eigenvalues is examined. Moreover, a second quantization of the resulting Schro¨dinger equation is performed, and a convergent solution for the nontrivial zeros of the analytic continuation of the Riemann zeta function is obtained. Finally, from the holomorphicity of the eigenvalues it is shown that the real part of every nontrivial zero of the Riemann zeta function converges at σ = 1/2.

Biased behavior of weighted Mertens sums

2019 ◽  
Vol 16 (03) ◽  
pp. 547-577
Author(s):  
Emre Alkan
Keyword(s):  
Zeta Function ◽  
Riemann Zeta Function ◽  
Riemann Hypothesis ◽  
Zeta Functions ◽  
Prime Numbers ◽  
Summation Formulas ◽  
The Riemann Zeta Function ◽  
Convexity Properties ◽  
Riemann Zeta ◽  
Derivatives Of

Using convexity properties of reciprocals of zeta functions, especially the reciprocal of the Riemann zeta function, we show that certain weighted Mertens sums are biased in favor of square-free integers with an odd number of prime factors. We study such type of bias for different ranges of the parameters and then consider generalizations to Mertens sums supported on semigroups of integers generated by relatively large subsets of prime numbers. We further obtain a wider range for the parameters both unconditionally and then conditionally on the Riemann Hypothesis. At the same time, we extend to certain semigroups, two classical summation formulas originating from the works of Landau concerning the behavior of derivatives of the reciprocal of the Riemann zeta function at [Formula: see text].

scholarly journals The Riemann Hypothesis is most likely true

2021 ◽  
Author(s):  
Frank Vega
Keyword(s):  
Zeta Function ◽  
Riemann Zeta Function ◽  
Riemann Hypothesis ◽  
Complex Numbers ◽  
Natural Numbers ◽  
Chebyshev Function ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
Sum Of Divisors

In mathematics, the Riemann hypothesis is a conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part $\frac{1}{2}$. The Riemann hypothesis belongs to the David Hilbert's list of 23 unsolved problems and it is one of the Clay Mathematics Institute's Millennium Prize Problems. The Robin criterion states that the Riemann hypothesis is true if and only if the inequality $\sigma(n)< e^{\gamma } \times n \times \log \log n$ holds for all natural numbers $n> 5040$, where $\sigma(x)$ is the sum-of-divisors function and $\gamma \approx 0.57721$ is the Euler-Mascheroni constant. The Nicolas criterion states that the Riemann hypothesis is true if and only if the inequality $\prod_{q \leq q_{n}} \frac{q}{q-1} > e^{\gamma} \times \log\theta(q_{n})$ is satisfied for all primes $q_{n}> 2$, where $\theta(x)$ is the Chebyshev function. Using both inequalities, we show that the Riemann hypothesis is most likely true.

scholarly journals FRACTAL SUPERSYMMETRIC QM, GEOMETRIC PROBABILITY AND THE RIEMANN HYPOTHESIS

2004 ◽  
Vol 01 (06) ◽  
pp. 751-793 ◽  
Author(s):  
CARLOS CASTRO ◽  
JORGE MAHECHA
Keyword(s):  
Imaginary Axis ◽  
Hermitian Operator ◽  
Theta Series ◽  
Geometric Probability ◽  
Weierstrass Function ◽  
Nontrivial Zeros ◽  
Hyperbolic Sine ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
The Relationship

The Riemann's hypothesis (RH) states that the nontrivial zeros of the Riemann zeta-function are of the form sn=1/2+iλn. Earlier work on the RH based on supersymmetric QM, whose potential was related to the Gauss–Jacobi theta series, allows us to provide the proper framework to construct the well-defined algorithm to compute the density of zeros in the critical line, which would complement the existing formulas in the literature for the density of zeros in the critical strip. Geometric probability theory furnishes the answer to the difficult question whether the probability that the RH is true is indeed equal to unity or not. To test the validity of this geometric probabilistic framework to compute the probability if the RH is true, we apply it directly to the the hyperbolic sine function sinh (s) case which obeys a trivial analog of the RH (the HSRH). Its zeros are equally spaced in the imaginary axis sn=0+inπ. The geometric probability to find a zero (and an infinity of zeros) in the imaginary axis is exactly unity. We proceed with a fractal supersymmetric quantum mechanical (SUSY-QM) model implementing the Hilbert–Polya proposal to prove the RH by postulating a Hermitian operator that reproduces all the λn for its spectrum. Quantum inverse scattering methods related to a fractal potential given by a Weierstrass function (continuous but nowhere differentiable) are applied to the fractal analog of the Comtet–Bandrauk–Campbell (CBC) formula in SUSY QM. It requires using suitable fractal derivatives and integrals of irrational order whose parameter β is one-half the fractal dimension (D=1.5) of the Weierstrass function. An ordinary SUSY-QM oscillator is also constructed whose eigenvalues are of the form λn=nπ and which coincide with the imaginary parts of the zeros of the function sinh (s). Finally, we discuss the relationship to the theory of 1/f noise.

scholarly journals The Riemann Hypothesis Is True

2021 ◽  
Author(s):  
Frank Vega
Keyword(s):  
Zeta Function ◽  
Riemann Zeta Function ◽  
Riemann Hypothesis ◽  
Complex Numbers ◽  
Natural Numbers ◽  
Chebyshev Function ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
Sum Of Divisors

The Riemann hypothesis is a conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part $\frac{1}{2}$. The Riemann hypothesis belongs to the David Hilbert's list of 23 unsolved problems. Besides, it is one of the Clay Mathematics Institute's Millennium Prize Problems. This problem has remained unsolved for many years. The Robin criterion states that the Riemann hypothesis is true if and only if the inequality $\sigma(n)< e^{\gamma } \times n \times \log \log n$ holds for all natural numbers $n>5040$, where $\sigma(x)$ is the sum-of-divisors function and $\gamma \approx 0.57721$ is the Euler-Mascheroni constant. The Nicolas criterion states that the Riemann hypothesis is true if and only if the inequality $\prod_{q \leq q_{n}} \frac{q}{q-1}>e^{\gamma} \times \log\theta(q_{n})$ is satisfied for all primes $q_{n}>2$, where $\theta(x)$ is the Chebyshev function. Using both inequalities, we show that the Riemann hypothesis is true.

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.

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