On the random nature of (prime) number distribution

Print ISBN: 978-93-48388-06-3, eBook ISBN: 978-93-48388-54-4, 2024

In this paper I present the distribution of prime numbers which was treated in many researches by studying the function of Riemann; because it has a remarkable property; its non trivial zeros are prime numbers; but in this work I will show that we can find the distribution of prime numbers on remaining in natural numbers onl

2021

In this research first, a sequence of properties called delta is assigned to each prime number and then examined. Deltas are only dependent on the distribution of prime numbers, so the results obtained for the delta distribution can be considered as a proxy for the distribution of prime numbers. The first observation was that these properties are not unique and different prime numbers may have the same value of delta of a given order. It was found that a small number of deltas cover a large portion of prime numbers, so by recognizing repetitive deltas, the next prime numbers can be predicted with a certain probability, but the most important observation of this study is the normal distribution of deltas. This research has not tried to justify the obtained observations and instead of answering the questions, it seeks to ask the right question.

2017

Currently there is no known efficient formula for primes. Besides that, prime numbers have great importance in e.g., information technology such as public-key cryptography, and their position and possible or impossible functional generation among the natural numbers is an ancient dilemma. The properties of the functions 2ab+a+b in the domain of natural numbers are introduced, analyzed, and exhibited to illustrate how these single out all the prime numbers from the full set of odd numbers. The characterization of odd primes vs. odd non-primes can be done with 2ab+a+b among the odd natural numbers as an analogue to the other, well known type of fundamental characterization for irrational and rational numbers among the real numbers. The prime number theorem, twin primes and erratic nature of primes, are also commented upon with respect to selection, as well as with the Fermat and Euler numbers as examples. Keywords prime number generator, prime number theorem, twin primes, erratic nature of primes

2017

Currently there is no known efficient formula for primes. Besides that, prime numbers have great importance in e.g., information technology such as public-key cryptography, and their position and possible or impossible functional generation among the natural numbers is an ancient dilemma. The properties of the functions 2ab+a+b in the domain of natural numbers are introduced, analyzed, and exhibited to illustrate how these single out all the prime numbers from the full set of odd numbers. The characterization of odd primes vs. odd non-primes can be done with 2ab+a+b among the odd natural numbers as an analogue to the other, well known type of fundamental characterization for irrational and rational numbers among the real numbers. The prime number theorem, twin primes and erratic nature of primes, are also commented upon with respect to selection, as well as with the Fermat and Euler numbers as examples.

2013

In this paper we discuss Legendre’s, Brocard’s, Andrica’s, and Oppermann’s conjectures. We propose a conjecture regarding the distribution of prime numbers and we also prove that if it is true, then the previously mentioned conjectures follow. Moreover, we also show that if the conjecture in question holds, then there is at least one prime number in the interval [n, n+ 2 b √ nc − 1] for every positive integer n.

In this talk, we give a panorama of the proof of the Riemann Hypothesis we have in recent years, see [?], which is based on the proofs of the strong density hypothesis in [?] and the strong Lindelöf hypothesis in [?]. The proof of RH is "short", but the preparation is quite a lengthy work. For a rough sketch of the proof of the Riemann Hypothesis in a short article instead of a panorama as in this talk, one may see [?]. We shall discuss five symmetries: the conjugate symmetry, algebra and analysis or the prime counting function and the zero-free region of the Riemann zeta function, the pseudogamma functions, various alternation of the Riemann zeta functions and the various prime counting functions like π(x), ψ(x), and ϖ(x), the almost symmetric routes, and, the Lambda function with log(p) when n is a prime powers vesus −1 otherwise in five subjects: applied mathematics, algebra, analysis, computer algebra, and topology.

arXiv: Number Theory, 2018

A numerical study on the distributions of primes in short intervals of length $h$ over the natural numbers $N$ is presented. Based on Cram\'er's model in Number Theory, we obtain a heuristic expression applicable when $h \gg \log{N}$ but $h \ll N$, providing support to the Montgomery and Soundararajan conjecture on the variance of the prime distribution at this scale.

2018

A numerical study on the distributions of primes in short intervals of length h over the natural numbers N is presented. Based on Cramér's model in Number Theory, we obtain a heuristic expression applicable when h log N but h N , providing support to the Montgomery and Soundararajan conjecture on the variance of the prime distribution at this scale.

Journal of Number Theory, 1987

We consider the problem of estimating the number ψ(x,x α ,x β) of integers in the interval (x − x β , x] having no prime factor greater than x α. We study when one can guarantee ψ(x,x α ,x β) > 0 for large x and when one can guarantee ψ(x,x α ,x β) ≥ c(α,β) x β for large x, for some positive constant c(α,β). In particular

2000