scholarly journals Relation between primes and nontrivial zeros in the Riemann hypothesis; Legendre polynomials, modified zeta function and Schrödinger equation

2012 ◽  
Vol 53 (12) ◽  
pp. 122108 ◽  
Author(s):  
Seongsoo Choi ◽  
J. W. Chung ◽  
Kwang S. Kim
Keyword(s):  
Schrödinger Equation ◽  
Zeta Function ◽  
Riemann Hypothesis ◽  
Legendre Polynomials ◽  
Schrodinger Equation ◽  
Nontrivial 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 Farey Sequences and the Riemann Hypothesis

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.

scholarly journals Farey Sequences and the Riemann Hypothesis

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.

scholarly journals Farey Sequences and the Riemann Hypothesis

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.

scholarly journals RIEMANN ZETA ZEROS AND PRIME NUMBER SPECTRA IN QUANTUM FIELD THEORY

2013 ◽  
Vol 28 (26) ◽  
pp. 1350128 ◽  
Author(s):  
G. MENEZES ◽  
B. F. SVAITER ◽  
N. F. SVAITER
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Zeta Function ◽  
Riemann Hypothesis ◽  
Adjoint Operator ◽  
Zeta Functions ◽  
Prime Numbers ◽  
Quantum Field ◽  
Nontrivial Zeros ◽  
Riemann Zeta

The Riemann hypothesis states that all nontrivial zeros of the zeta function lie in the critical line Re (s) = 1/2. Hilbert and Pólya suggested that one possible way to prove the Riemann hypothesis is to interpret the nontrivial zeros in the light of spectral theory. Using the construction of the so-called super-zeta functions or secondary zeta functions built over the Riemann nontrivial zeros and the regularity property of one of this function at the origin, we show that it is possible to extend the Hilbert–Pólya conjecture to systems with countably infinite number of degrees of freedom. The sequence of the nontrivial zeros of the Riemann zeta function can be interpreted as the spectrum of a self-adjoint operator of some hypothetical system described by the functional approach to quantum field theory. However, if one considers the same situation with numerical sequences whose asymptotic distributions are not "far away" from the asymptotic distribution of prime numbers, the associated functional integral cannot be constructed. Finally, we discuss possible relations between the asymptotic behavior of a sequence and the analytic domain of the associated zeta function.

The Riemann hypothesis and universality of the Riemann zeta-function

2018 ◽  
Vol 68 (4) ◽  
pp. 741-748 ◽  
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).

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

2016 ◽  
Vol 162 (2) ◽  
pp. 293-317 ◽  
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.

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.

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