Power spectrum of the difference between the prime-number counting function and Riemann's function: 1/f2?

2004 ◽  
Vol 334 (3-4) ◽  
pp. 477-481 ◽  
Author(s):  
Boon Leong Lan ◽  
Shaohen Yong
Keyword(s):  
Power Spectrum ◽  
Prime Number ◽  
Counting Function ◽  
The Difference

scholarly journals A Proof of the Goldbach and Polignac Conjectures

2020 ◽  
Author(s):  
Jason R. South
Keyword(s):  
Prime Number ◽  
The Difference

For any prime $p_i \leq a$, where $a \in \mathbb{N}$ and $a > 3$, it is possible to define some $b_i \in \mathbb{N}$ where $a - b_i = p_i$. Using this fact, it will be shown that there exists some $q_i \in \mathbb{N}$ where $a + b_i = q_i$, and $q_i$ is a prime number, by proving solutions cannot exist when $2a = q_i +p_i$ for all prime $p_i \leq a$, and $\prod_{i = 1}^{\pi(a)}q_i = \prod_{i = 1}^{\pi(a)}p_i^{\alpha_i}$ for some $\alpha_i \in \mathbb{N}$ when $a > 3$. A similar method will be employed to prove every even number is the difference of two primes by assigning to any odd, prime $p_i \leq a$ some $b_i' \in \mathbb{N}$ where $a - b_i' = -p_i$, along with the existence of $q_i' \in \mathbb{N}$ where $a +b_i' = q_i'$ and showing solutions cannot exist for $a > 3$ when $2a = q_i' - p_i$ for all odd $p_i \leq a$, and $\prod_{i = 2}^{\pi(a)}q_i' = \prod_{i = 2}^{\pi(a)}p_i^{\alpha_i'}$ for some $\alpha_i' \in \mathbb{N}$. The Polignac Conjecture will then follow.

scholarly journals Proof the Skewes’ number is not an integer using lattice points and tangent line

2021 ◽  
Vol 17 (2) ◽  
pp. 5-18
Author(s):  
V. Ďuriš ◽  
T. Šumný ◽  
T. Lengyelfalusy
Keyword(s):  
Counting Function ◽  
Prime Number Theorem ◽  
Prime Numbers ◽  
Integral Function ◽  
Lattice Points ◽  
Tangent Line ◽  
Miller Test ◽  
Logarithmic Integral ◽  
The Difference ◽  
Prime Counting Function

Abstract Skewes’ number was discovered in 1933 by South African mathematician Stanley Skewes as upper bound for the first sign change of the difference π (x) − li(x). Whether a Skewes’ number is an integer is an open problem of Number Theory. Assuming Schanuel’s conjecture, it can be shown that Skewes’ number is transcendental. In our paper we have chosen a different approach to prove Skewes’ number is an integer, using lattice points and tangent line. In the paper we acquaint the reader also with prime numbers and their use in RSA coding, we present the primary algorithms Lehmann test and Rabin-Miller test for determining the prime numbers, we introduce the Prime Number Theorem and define the prime-counting function and logarithmic integral function and show their relation.

scholarly journals Development of sieve of Eratosthenes and sieve of Sundaram's proof

2021 ◽  
Author(s):  
Ahmed Hamdy Diab
Keyword(s):  
Prime Number ◽  
Prime Numbers ◽  
Run Time ◽  
The Difference

Abstract We make two algorithms that generate all prime numbers up to a given limit, they are a development of sieve of Eratosthenes algorithm, we use two formulas to achieve this development, where all the multiples of prime number 2 are eliminated in the first formula, and all the multiples of prime numbers 2 and 3 are eliminated in the second formula. Using the first algorithm we proof sieve of Sundaram's algorithm, then we improve it to be more efficient prime generating algorithm. We will show the difference in performance between all the algorithms we will make and sieve of Eratosthenes algorithm in terms of run time.

scholarly journals Pretentious multiplicative functions and the prime number theorem for arithmetic progressions

2013 ◽  
Vol 149 (7) ◽  
pp. 1129-1149 ◽  
Author(s):  
Dimitris Koukoulopoulos
Keyword(s):  
Prime Number ◽  
Elementary Proof ◽  
Counting Function ◽  
Prime Number Theorem ◽  
Arithmetic Progressions ◽  
Multiplicative Functions ◽  
Primes In Arithmetic Progressions

AbstractBuilding on the concept ofpretentious multiplicative functions, we give a new and largely elementary proof of the best result known on the counting function of primes in arithmetic progressions.

scholarly journals LINE ASYMMETRY AND EXCITATION MECHANISM OF SOLAR OSCILLATIONS

1998 ◽  
Vol 185 ◽  
pp. 195-198
Author(s):  
R. Nigam ◽  
A.G. Kosovichev ◽  
P.H. Scherrer ◽  
J. Schou
Keyword(s):  
Power Spectrum ◽  
Power Spectra ◽  
Theoretical Explanation ◽  
Solar Oscillations ◽  
Localized Source ◽  
Line Asymmetry ◽  
The Difference ◽  
First Time ◽  
Opening Address

In his opening address at the conference Dr. Tim Brown posed the line asymmetry problem between velocity and intensity as a puzzle in helioseismology that has been resisting theoretical explanation for many years. It was the observations of Duvall et al. (1993) that for the first time indicated that the power spectrum of solar acoustic modes show varying amounts of asymmetry. In particular, the velocity and intensity power spectra revealed an opposite sense of asymmetry. Many doubted the correctness of the experiment and thought it to be a puzzling result (Abrams & Kumar, 1996). Many authors have investigated this problem theoretically and have found that there is an inherent asymmetry whenever there is a localized source exciting the solar oscillations (Gabriel, 1995; Roxburgh & Vorontsov, 1995; Abrams & Kumar, 1996; Nigam et al. 1997). This problem has important implications in helioseismology where the eigenfrequencies are generally determined by assuming that the power spectrum was symmetric and can be fitted by a Lorentzian. This leads to systematic errors in the determination of frequencies and, thus, affects the results of inversions (Rhodes et al. 1997). In this paper we offer an explanation for the difference in parity of the two asymmetries and estimate the depth and type of the sources that are responsible for exciting the solar p-modes.

Two ranges in blood pressure power spectrum with different 1/f characteristics

1994 ◽  
Vol 267 (2) ◽  
pp. H449-H454 ◽  
Author(s):  
C. D. Wagner ◽  
P. B. Persson
Keyword(s):  
Blood Pressure ◽  
Time Series ◽  
Power Spectrum ◽  
Low Frequency ◽  
Frequency Range ◽  
F Region ◽  
Autonomic Blockade ◽  
Beta Receptor Blockade ◽  
The Difference ◽  
Fast Fourier Analysis

Most time series of biological systems contain a considerable amount of 1/f noise. This form of noise is characterized by fluctuations in which power steadily increases at lower frequencies. To determine the origin of 1/f noise, blood pressure (BP) was measured over 4 h in conscious foxhounds. The power spectrum of BP was obtained by fast Fourier analysis. After log-log transformation, the power spectrum (log power vs. log frequency) characteristically revealed a linear regression. Surprisingly, there were two 1/f ranges. The first 1/f region was located within a low-frequency range (< 10(-1.7) Hz; slope -0.9; r = -0.9). The second 1/f range was identified at 10(-1.4) to 10(-1) Hz (slope -1.2; r = -0.7). After baroreceptor denervation (n = 7), the steepness of both slopes increased significantly (P < 0.05 for lower 1/f range, P < 0.001 for higher 1/f range), and the difference in slopes was clearly greater (slope in lower range -1.2; r = 0.96 vs. -3.1, r = -0.92 in the higher range; P < 0.001). Neither alpha-receptor (n = 6) nor beta-receptor blockade (n = 4) considerably changed the slopes after denervation. However, autonomic blockade (n = 5) restored the slope in the low-frequency range (-0.9; r = -0.9). In conclusion, there are two independently modulated 1/f frequency ranges in BP time series. Baroreceptors especially attenuate 1/f noise in the higher frequency range.

scholarly journals Combinatorial Models of the Distribution of Prime Numbers

Mathematics ◽  
2021 ◽  
Vol 9 (11) ◽  
pp. 1224
Author(s):  
Vito Barbarani
Keyword(s):  
Prime Number ◽  
Counting Function ◽  
Stirling Numbers ◽  
Prime Number Theorem ◽  
Prime Numbers ◽  
Lower And Upper Bounds ◽  
Integer Sequences ◽  
New Class ◽  
Random Integer ◽  
Prime Counting Function

This work is divided into two parts. In the first one, the combinatorics of a new class of randomly generated objects, exhibiting the same properties as the distribution of prime numbers, is solved and the probability distribution of the combinatorial counterpart of the n-th prime number is derived together with an estimate of the prime-counting function π(x). A proposition equivalent to the Prime Number Theorem (PNT) is proved to hold, while the equivalent of the Riemann Hypothesis (RH) is proved to be false with probability 1 (w.p. 1) for this model. Many identities involving Stirling numbers of the second kind and harmonic numbers are found, some of which appear to be new. The second part is dedicated to generalizing the model to investigate the conditions enabling both PNT and RH. A model representing a general class of random integer sequences is found, for which RH holds w.p. 1. The prediction of the number of consecutive prime pairs as a function of the gap d, is derived from this class of models and the results are in agreement with empirical data for large gaps. A heuristic version of the model, directly related to the sequence of primes, is discussed, and new integral lower and upper bounds of π(x) are found.

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