scholarly journals Possibility to Construct a Machine for Primality Testing of Numbers

2021 ◽  
pp. 21-29
Author(s):  
Takaaki Musha
Keyword(s):  
Correlation Function ◽  
Theoretical Analysis ◽  
Zeta Function ◽  
Prime Number ◽  
White Light ◽  
Riemann Zeta Function ◽  
Single Spectrum ◽  
Primality Testing ◽  
Optical Prism ◽  
Riemann Zeta

Like the optical prism to break white light up into its constituent spectral colors, the machine to show a prime as a single spectrum is proposed. From the theoretical analysis, it can be shown that the machine to recognize the prime number as a single spectrum can be realized by using the correlation function of Riemann zeta function. Moreover, this method can be used for a factorization of the integer consisted of two primes.

scholarly journals HIGHER CORRELATIONS AND THE ALTERNATIVE HYPOTHESIS

2020 ◽  
Vol 71 (1) ◽  
pp. 257-280
Author(s):  
Jeffrey C Lagarias ◽  
Brad Rodgers
Keyword(s):  
Correlation Function ◽  
Zeta Function ◽  
Point Process ◽  
Riemann Zeta Function ◽  
Alternative Hypothesis ◽  
Pair Correlation ◽  
Local Statistics ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
Rule Out

Abstract The Alternative Hypothesis (AH) concerns a hypothetical and unlikely picture of how zeros of the Riemann zeta function are spaced, which one would like to rule out. In the Alternative Hypothesis, the renormalized distance between non-trivial zeros is supposed to always lie at a half integer. It is known that the Alternative Hypothesis is compatible with what is known about the pair correlation function of zeta zeros. We ask whether what is currently known about higher correlation functions of the zeros is sufficient to rule out the Alternative Hypothesis and show by construction of an explicit counterexample point process that it is not. A similar result was recently independently obtained by Tao, using slightly different methods. We also apply the ergodic theorem to this point process to show there exists a deterministic collection of points lying in $\tfrac{1}{2}\mathbb{Z}$, which satisfy the Alternative Hypothesis spacing, but mimic the local statistics that are currently known about zeros of the zeta function.

scholarly journals The Nicolas criterion for the Riemann Hypothesis

2021 ◽  
Author(s):  
Frank Vega
Keyword(s):  
Zeta Function ◽  
Prime Number ◽  
Riemann Zeta Function ◽  
Riemann Hypothesis ◽  
Complex Numbers ◽  
Chebyshev Function ◽  
The Riemann Zeta Function ◽  
Riemann Zeta

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 $\frac{1}{2}$. For every prime number $p_{n}$, we define the sequence $X_{n} = \prod_{q \leq p_{n}} \frac{q}{q-1} - e^{\gamma} \times \log \theta(p_{n})$, where $\theta(x)$ is the Chebyshev 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 $X_{n} > 0$ holds for all primes $p_{n} > 2$. For every prime number $p_{k} > 2$, $X_{k} > 0$ is called the Nicolas inequality. We prove that the Nicolas inequality holds for all primes $p_{n} > 2$. In this way, we demonstrate that the Riemann hypothesis is true.

Summability by Dirichlet convolutions

1967 ◽  
Vol 63 (2) ◽  
pp. 393-400 ◽  
Author(s):  
S. L. Segal
Keyword(s):  
Zeta Function ◽  
Prime Number ◽  
Riemann Zeta Function ◽  
Prime Number Theorem ◽  
Summation Method ◽  
Cesàro Means ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
Image Position ◽  
Equivalent Method

Ingham (3) discusses the following summation method:A series ∑an will be said to be summable to s ifwhere, as usual, [x] indicates the greatest integer ≤ x. (An equivalent method was introduced somewhat earlier by Wintner (8), but the notation (I) for the above method and the attachment to Ingham's name seem to have become usual following [(1), Appendix IV].) The method (I) is intimately connected with the prime number theorem and the fact that the Riemann zeta-function ζ(s) has no zeros on the line σ = 1. Ingham proved, among other results, that (I) is not comparable with convergence but, nevertheless, for every δ > 0, (I) ⇒ (C, δ) and for every δ, 0 < δ < 1, (C, −δ) ⇒ (I), where the (C, k) are Cesàro means of order k.

scholarly journals The Nicolas Criterion for the Riemann Hypothesis

2021 ◽  
Author(s):  
Frank Vega
Keyword(s):  
Zeta Function ◽  
Prime Number ◽  
Riemann Zeta Function ◽  
Riemann Hypothesis ◽  
Complex Numbers ◽  
Chebyshev Function ◽  
The Riemann Zeta Function ◽  
Riemann Zeta

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 $\frac{1}{2}$. For every prime number $p_{n}$, we define the sequence $X_{n} = \prod_{q \leq p_{n}} \frac{q}{q-1} - e^{\gamma} \times \log \theta(p_{n})$, where $\theta(x)$ is the Chebyshev 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 $X_{n} > 0$ holds for all primes $p_{n} > 2$. For every prime number $p_{k} > 2$, $X_{k} > 0$ is called the Nicolas inequality. We prove that the Nicolas inequality holds for all primes $p_{n} > 2$. In this way, we demonstrate that the Riemann hypothesis is true.

A Tauberian Theorem and Analogues of the Prime Number Theorem

1968 ◽  
Vol 20 ◽  
pp. 362-367 ◽  
Author(s):  
T. M. K. Davison
Keyword(s):  
Zeta Function ◽  
Prime Number ◽  
Riemann Zeta Function ◽  
Tauberian Theorem ◽  
Prime Number Theorem ◽  
Negative Function ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
Image Position

In 1945 Ingham (3) proved the following Tauberian theorem: if ƒ is a non-decreasing, non-negative function on [1, ∞) and1then ƒ(x) ∼ cx. His proof is based on the non-vanishing of the Riemann zeta-function, ζ (s), on the line , and uses Pitt's form of Wiener's Tauberian theorem; (see, e.g., 5, Theorem 109, p. 211).

Sign in / Sign up

Export Citation Format

Share Document