THE RIEMANN HYPOTHESIS IS A CONSEQUENCE OF $\mathcal{CT}$-INVARIANT QUANTUM MECHANICS

The Riemann hypothesis (RH) states that the non-trivial zeros of the Riemann zeta-function are of the form sn = 1/2+iλn. An improvement of our previous construction to prove the RH is presented by implementing the Hilbert–Polya proposal and furnishing the Fractal Supersymmetric Quantum Mechanical (SUSY-QM) model whose spectrum reproduces the imaginary parts of the zeta zeros. We model the fractal fluctuations of the smooth Wu–Sprung potential (that capture the average level density of zeros) by recurring to a weighted superposition of Weierstrass functions ∑p W(x, p, D) and where the summation has to be performed over all primes p in order to recapture the connection between the distribution of zeta zeros and prime numbers. We proceed next with the construction of a smooth version of the fractal QM wave equation by writing an ordinary Schroedinger equation whose fluctuating potential (relative to the smooth Wu–Sprung potential) has the same functional form as the fluctuating part of the level density of zeros. The second approach to prove the RH relies on the existence of a continuous family of scaling-like operators involving the Gauss–Jacobi theta series. An explicit completion relation ("trace formula") related to a superposition of eigenfunctions of these scaling-like operators is defined. If the completion relation is satisfied, this could be another test of the Riemann Hypothesis. In an appendix, we briefly describe our recent findings showing why the Riemann Hypothesis is a consequence of [Formula: see text]-invariant Quantum Mechanics, because [Formula: see text] where s are the complex eigenvalues of the scaling-like operators. We show why [Formula: see text] invariance requires that s(1 - s) = real , which implies that s is real and/or it lies in the critical Riemann line.

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FRACTAL SUPERSYMMETRIC QM, GEOMETRIC PROBABILITY AND THE RIEMANN HYPOTHESIS

International Journal of Geometric Methods in Modern Physics ◽

10.1142/s0219887804000393 ◽

2004 ◽

Vol 01 (06) ◽

pp. 751-793 ◽

Cited By ~ 6

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.

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

Symmetry ◽

10.3390/sym13122410 ◽

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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Farey Sequences and the Riemann Hypothesis

10.20944/preprints201607.0003.v5 ◽

2016 ◽

Author(s):

Darrell Cox

Keyword(s):

Zeta Function ◽

Riemann Hypothesis ◽

Analytic Number Theory ◽

Square Root ◽

Nontrivial Zeros ◽

Landau Theorem ◽

Farey Sequences ◽

The Riemann Zeta Function ◽

Riemann Zeta ◽

Function Zeros

Relationships between the Farey sequence and the Riemann hypothesis other than the Franel-Landau theorem are discussed. Whether a function similar to Chebyshev’s second function is square-root close to a line having a slope different from 1 is discussed. The nontrivial zeros of the Riemann zeta function can be used to approximate many functions in analytic number theory. For example, it could be said that the nontrival zeta function zeros and the Möbius function generate in essence the same function - the Mertens function. A different approach is to start with a sequence that is analogous to the nontrivial zeros of the zeta function and follow the same procedure with both this sequence and the nontrivial zeros of the zeta function to generate in essence the same function. A procedure for generating such a function is given.

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Farey Sequences and the Riemann Hypothesis

10.20944/preprints201607.0003.v3 ◽

2016 ◽

Author(s):

Darrell Cox

Keyword(s):

Zeta Function ◽

Riemann Hypothesis ◽

Analytic Number Theory ◽

Square Root ◽

Nontrivial Zeros ◽

Landau Theorem ◽

Farey Sequences ◽

The Riemann Zeta Function ◽

Riemann Zeta ◽

Function Zeros

Relationships between the Farey sequence and the Riemann hypothesis other than the Franel-Landau theorem are discussed. Whether a function similar to Chebyshev’s second function is square-root close to a line having a slope different from 1 is discussed. The nontrivial zeros of the Riemann zeta function can be used to approximate many functions in analytic number theory. For example, it could be said that the nontrival zeta function zeros and the Möbius function generate in essence the same function - the Mertens function. A different approach is to start with a sequence that is analogous to the nontrivial zeros of the zeta function and follow the same procedure with both this sequence and the nontrivial zeros of the zeta function to generate in essence the same function. A procedure for generating such a function is given.

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Farey Sequences and the Riemann Hypothesis

10.20944/preprints201607.0003.v1 ◽

2016 ◽

Author(s):

Darrell Cox

Keyword(s):

Zeta Function ◽

Riemann Hypothesis ◽

Analytic Number Theory ◽

Square Root ◽

Nontrivial Zeros ◽

Landau Theorem ◽

Farey Sequences ◽

The Riemann Zeta Function ◽

Riemann Zeta ◽

Function Zeros

Relationships between the Farey sequence and the Riemann hypothesis other than the Franel-Landau theorem are discussed. Whether a function similar to Chebyshev’s second function is square-root close to a line having a slope different from 1 is discussed. The nontrivial zeros of the Riemann zeta function can be used to approximate many functions in analytic number theory. For example, it could be said that the nontrival zeta function zeros and the Möbius function generate in essence the same function - the Mertens function. A different approach is to start with a sequence that is analogous to the nontrivial zeros of the zeta function and follow the same procedure with both this sequence and the nontrivial zeros of the zeta function to generate in essence the same function. A procedure for generating such a function is given.

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The Riemann hypothesis and universality of the Riemann zeta-function

Mathematica Slovaca ◽

10.1515/ms-2017-0141 ◽

2018 ◽

Vol 68 (4) ◽

pp. 741-748 ◽

Cited By ~ 4

Author(s):

Ramūnas Garunkštis ◽

Antanas Laurinčikas

Keyword(s):

Zeta Function ◽

Wide Class ◽

Riemann Zeta Function ◽

Analytic Functions ◽

Riemann Hypothesis ◽

Nontrivial Zeros ◽

The Riemann Zeta Function ◽

Riemann Zeta

Abstract We prove that, under the Riemann hypothesis, a wide class of analytic functions can be approximated by shifts ζ(s + iγk), k ∈ ℕ, of the Riemann zeta-function, where γk are imaginary parts of nontrivial zeros of ζ(s).

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ON THE EXPLICIT FORMULA IN THE THEORY OF PRIME NUMBERS

International Journal of Number Theory ◽

10.1142/s1793042112500327 ◽

2012 ◽

Vol 08 (03) ◽

pp. 589-597 ◽

Cited By ~ 1

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.

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The distribution of k-free numbers and the derivative of the Riemann zeta-function

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004116000554 ◽

2016 ◽

Vol 162 (2) ◽

pp. 293-317 ◽

Cited By ~ 1

Author(s):

XIANCHANG MENG

Keyword(s):

Characteristic Function ◽

Zeta Function ◽

Riemann Zeta Function ◽

Riemann Hypothesis ◽

Limiting Distribution ◽

Maximum Order ◽

Nontrivial Zeros ◽

The Riemann Zeta Function ◽

Riemann Zeta ◽

Heuristic Analysis

AbstractUnder the Riemann Hypothesis, we connect the distribution of k-free numbers with the derivative of the Riemann zeta-function at nontrivial zeros of ζ(s). Moreover, with additional assumptions, we prove the existence of a limiting distribution of $e^{-\frac{y}{2k}}M_k(e^y)$ and study the tail of the limiting distribution, where $M_k(x)=\sum_{n\leq x}\mu_k(n)-{x}/{\zeta(k)}$ and μk(n) is the characteristic function of k-free numbers. Finally, we make a conjecture about the maximum order of Mk(x) by heuristic analysis on the tail of the limiting distribution.

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A Schrödinger Equation for Solving the Riemann Hypothesis

10.20944/preprints201712.0149.v2 ◽

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.

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