The prime number theorem via Fourier transformation

During the analysis of the fractal-primorial periodicity of the natural series of numbers, presented in the form of an alternation (sequence) of prime numbers (1 smallest prime factor > 1 of any integer), the regularity of prime numbers distribution was revealed. That is, the theorem is proved that for any integer = N on the segment of the natural series of numbers from 1 to N + 2N: (1) prime numbers are arranged in groups, by exactly three consecutive prime numbers of the form: (Р1-Р2-Р3). In this case, the distance from the first to the third prime number of any group is less than 2N integers, that is, Р3–Р1 < 2N integers. (2) These same prime numbers are redistributed in a line in groups, by exactly two consecutive prime numbers, on all segments of the natural series of numbers shorter than 2Nintegers.

It is practically impossible to break a cryptosystem like RSA if the key is large enough with the prime numbers are randomly distributed. Recent research shows prime numbers show a certain distribution pattern. This makes it practically possible to break such systems by solving the factoring problem. By computing the distribution probabilities and finite field incremental changes of connected prime numbers, we have validated Robert J. Lemke Oliver and Kannan Soundararajan's study findings on trailing prime numbers. We proposed and developed ranked selection algorithm on prime number distribution. We developed a number of applications, including proposing one for fast factoring the product of two large prime numbers, also known as the factoring problem.

The Riesz means, or sometimes typical means, were introduced by M. Riesz and have been studied in connection with summability of Fourier series and of Dirichlet series [8] and [11]. In number-theoretic context, it is the Riesz sum rather than the Riesz mean that has been extensively studied. The Riesz sums appear as long as there appears the G-function. Cf. Remark 1 and [14]. As is shown below, the Riesz sum corresponds to integration while Landau's differencing is an analogue of differentiation. This integration-differentiation aspect has been the driving force of many researches on number-theoretic asymptotic formulas. Ingham's decent treatment [13] of the prime number theorem is one typical example. We state some efficient theorems that give asymptotic formulas for the sums of coefficients of the generating Dirichlet series not necessarily satisfying the functional equation.

In this study, sequences are used for direct computation of prime numbers. A single equation for generating all prime numbers with the exception of 2 and 3 and composite numbers that are not divisible by 2 or 3 is presented. A periodicity to the indices of the composites generated by that equation is determined. An equation to determine the indices of the composite numbers is derived. The equation for determining the composite indices is then altered to avoid redundancy by observation of its diagonal, when its values are inserted into a matrix that has the indexes (j) for columns and (k) for rows. The two equations are presented in several different forms and are used in conjunction to render a technique for computing prime numbers indefinitely. The validity of the technique is proven up to one billion by computation and the prime number theorem.

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

Prime numbers are an important research area with application in computer security. The security of certain cryptographic schemes critical in information security lies in the difficulty of factoring large numbers into their prime factors. Determination of the distribution of prime numbers is an important unsolved problem in number theory. Biomolecules such as DNA have extended the domains of what is considered a standard computer model. This report investigated the distribution of prime numbers generated from viral DNA. The digits of Euler’s number and Pi were found to be encoded between Bacteriophage T4 Watson-Crick DNA segments comprised of prime numbers. Key words: Prime number, number theory, molecular computing, phage T4.

We give a quick tour through some topics in analytic prime number theory, focusing in particular on the strange mixture of order and chaos in the primes. For instance, while primes do obey some obvious patterns (e.g. they are almost all odd), and have a very regular asymptotic distribution (the prime number theorem), we still do not know a deterministic formula to quickly generate large numbers guaranteed to be prime, or to count even very simple patterns in the primes, such as twin primes p,p+2. Nevertheless, it is still possible in some cases to understand enough of the structure and randomness of the primes to obtain some quite nontrivial results.

The study of the primes is one of the most important branches of number theory. The only other branch of similar importance is Diophantine equations and this will be considered in the final three chapters. The literature on the prime numbers is very extensive beginning with Pythagoras and Euclid; even so, many problems remain. There are two pre-eminent results: Dirichlet’s theorem and the prime number theorem (PNT).

We discuss the formalization, in the Matita Interactive Theorem Prover, of some results by Chebyshev concerning the distribution of prime numbers, subsuming, as a corollary, Bertrand's postulate. Even if Chebyshev's result has been later superseded by the stronger prime number theorem, his machinery, and in particular the two functions psi and theta still play a central role in the modern development of number theory. The proof makes use of most part of the machinery of elementary arithmetics, and in particular of properties of prime numbers, gcd, products and summations, providing a natural benchmark for assessing the actual development of the arithmetical knowledge base.

In this paper it is proposed and proved an exact formula for the prime-counting function, finding an expression of Legendre's formula. As corollaries, they are proved some important conjectures regarding prime numbers distribution.

Suppose fi, f2, -*, fk are polynomials in one variable with all coefficients integral and leading coefficients positive, their degrees being hi, h2, **. , hk respectively. Suppose each of these polynomials is irreducible over the field of rational numbers and no two of them differ by a constant factor. Let Q(fi , f2, ... , fk ; N) denote the number of positive integers n between 1 and IV inclusive such that fi(n) , f2(n), , fk(n) are all primes. (We ignore the finitely many values of n for which some fi(n) is negative.) Then heuristically we would expect to have for N large

The highly irregular and rough fluctuations of the number of primes less or equal to a positive integer x for smaller values of x (x≤20,000) renders the approximations through the Prime Number Theorem quite unreliable. A fractal probability distribution more specifically, a multi-fractal fit to the density of primes less or equal to x for small values of x, is tried in this study. Results reveal that the multi-fractal fit to the density of primes in this situation outperforms the Prime Number Theorem approximation by almost 200% viz. the prediction error incurred by using the PNT approximation is double that of the multi-fractal fit to the density of primes. The study strongly suggests that a better multi-fractal distribution exists, even for large x, than the Prime Number approximation to the density of primes.

This paper provides new cases of the prime number theorem for Beurling’s generalized prime numbers. Let N be the distribution of a generalized number system and let π be the distribution of its primes. It is shown that N(x) = ax + O(x/ log x) (C), γ > 3/2, where (C) stands for the Cesaro sense, is sufficient for the prime number theorem to hold, i.e., π(x) ∼ x/ log x. The Cesaro asymptotic estimate explicitly means that ∫ x 1 N(t)− at t ( 1− t x )m dt = O ( x log x ) , for somem ∈ N. Therefore, it includes Beurling’s classical condition. We also show that under these conditions the Mobius function, associated to the generalized number system, has mean value equal to 0. The methods of this article are based on complex Tauberian theorems for local pseudo-function boundary behavior and arguments from the theory of asymptotic behavior of Schwartz distributions.

This textbook covers with the main topics in number theory as taught in universities throughout the world. Number theory deals mainly with properties of integers and rational numbers; it is not an organized theory in the usual sense, but a vast collection of individual topics and results, with some coherent sub-theories and a long list of unsolved problems. Nine main areas of the subject are treated: (i) divisibility and multiplicative functions; (ii) congruences and quadratic residues; (iii) the basics of algebraic numbers and sums of squares; (iv) continued fractions, diophantine approximations and transcendence; (v) quadratic forms; (vi) partitions; (vii) the prime numbers; (viii) diophantine equations; and (ix) elliptic curves (this part has been considerably extended for this edition). The increased use of computers in number theory is reflected in many sections (with much greater emphasis in this edition). Some results of a more advanced nature are also given, including the Gelfond-Schneider theorem, the prime number theorem, and the Mordell-Weil theorem. The latest work on fermat's last theorem is also breifly discussed. Each chapter ends with a collection of problems; hints or sketch solutions are given at the end of the book, together with various useful tables.

In a 2009 article, Barnett and Broughan considered the set of prime-index primes. If the prime numbers are listed in increasing order (2, 3, 5, 7, 11, 13, 17, . . .), then the prime-index primes are those which occur in a prime-numbered position in the list (3, 5, 11, 17, . . .). Barnett and Broughan established a prime-indexed prime number theorem analogous to the standard prime number theorem and gave an asymptotic for the size of the n-th prime-indexed prime. We give explicit upper and lower bounds for π2(x), the number of prime-indexed primes up to x, as well as upper and lower bounds on the n-th prime-indexed prime, all improvements on the bounds from 2009. We also prove analogous results for higher iterates of the sequence of primes. We present empirical results on large gaps between prime-index primes, the sum of inverses of the prime-index primes, and an analog of Goldbach’s conjecture for prime-index primes.