Prime Number Theorem Research Articles

A scaling property of Farey fractions. Part II: convergence at rational points

The Farey sequence of order n consists of all reduced fractions a / b between 0 and 1 with positive denominator b less or equal to n. The sums of the inverse denominators 1 / b of the Farey fractions in prescribed intervals with rational bounds have a simple main term, but the deviations are determined by an interesting sequence of polygonal functions \(f_n\). In a former paper we obtained a limit function for \(n \rightarrow \infty \), which describes a scaling behaviour of the functions \(f_n\) in the vicinity of any fixed rational number a / b and which is independent of a / b. In this paper we prove that \(f_n(a/b)\) tends to zero for \(n \rightarrow \infty \) by using elementary representation formulas for \(f_n(a/b)\) as well as a variant of the prime number theorem. An application of this result immediately gives a global version of the scaling behaviour of the functions \(f_n\) around the rational numbers.

A proof of Bertrand's postulate

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 Even if Chebyshev's result has been later superseded by the stronger prime number theorem, his machinery, and in particular the two functions psi and theta still play a central role in the modern development of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.

A variant of the Bombieri–Vinogradov theorem in short intervals and some questions of Serre

AbstractWe generalise the classical Bombieri–Vinogradov theorem for short intervals to a non-abelian setting. This leads to variants of the prime number theorem for short intervals where the primes lie in arithmetic progressions that are “twisted” by a splitting condition in a Galois extension of number fields. Using this result in conjunction with the recent work of Maynard, we prove that rational primes with a given splitting condition in a Galois extensionL/$\mathbb{Q}$exhibit bounded gaps in short intervals. We explore several arithmetic applications related to questions of Serre regarding the non-vanishing Fourier coefficients of cuspidal modular forms. One such application is that for a given modularL-functionL(s, f), the fundamental discriminantsdfor which thed-quadratic twist ofL(s, f) has a non-vanishing central critical value exhibit bounded gaps in short intervals.

Explicit formulas of a generalized Ramanujan sum

In this paper, explicit formulas involving a generalized Ramanujan sum are derived. An analogue of the prime number theorem is obtained and equivalences of the Riemann hypothesis are shown. Finally, explicit formulas of Bartz are generalized.

Products of binomial coefficients and unreduced Farey fractions

This paper studies the product [Formula: see text] of the binomial coefficients in the [Formula: see text]th row of Pascal’s triangle, which equals the reciprocal of the product of all the reduced and unreduced Farey fractions of order [Formula: see text]. It studies its size as a real number, measured by [Formula: see text], and its prime factorization, measured by the order of divisibility [Formula: see text] by a fixed prime [Formula: see text], each viewed as a function of [Formula: see text]. It derives three formulas for [Formula: see text], two of which relate it to base [Formula: see text] radix expansions of integers up to [Formula: see text], and which display different facets of its behavior. These formulas are used to determine the maximal growth rate of each [Formula: see text] and to explain structure of the fluctuations of these functions. It also defines analogous functions [Formula: see text] for all integer bases [Formula: see text] using base [Formula: see text] radix expansions replacing base [Formula: see text]-expansions. A final topic relates factorizations of [Formula: see text] to Chebyshev-type prime-counting estimates and the prime number theorem.

On Karamata’s proof of the Landau–Ingham Tauberian theorem

This is a self-contained exposition of (a generalization of) Karamata’s little known elementary proof of the Landau–Ingham Tauberian theorem, a result in real analysis from which the Prime Number theorem follows in a few lines.

An extension of the pair-correlation conjecture and applications

We study an extension of Montgomery's pair-correlation conjecture and its relevance in some problems on the distribution of prime numbers.

Polignac's Conjecture with New Prime Number Theorem

There are infinitely many pairs of consecutive primes which differ by even number En.Let Po(N, En) be the number of Polignac Prime Pairs (which difference by the even integer En) less than an integer (N+En), Pei be taken over the odd prime divisors of the even integer En less than √(N+En), Pni be taken over the odd primes less than √(N+En) except Pei, Pi be taken over the odd primes less than √(N+En), then exists the formulas as follows: Po(N, En) ≥ INT {N × (1-1/2) × Π (1-1/Pei) × Π (1-2/Pni)} - 1 ≥ INT {Ctwin × Ke(N) × 2N/(Ln (N+En))^2} - 1 Po(N, 2) ≥ INT {0.660 × 1.000 × 2N/(Ln (N+2))^2} - 1 Π (Pi(Pi-2)/(Pi-1)^2) ≥ Ctwin=0.6601618158… Ke(N)=Π( (1-1/Pei)/(1-2/Pei))=Π( (Pei-1)/(Pei-2)) ≥ 1 where -1 is except the natural integer 1.

Some examples in the theory of Beurling’s generalized prime numbers

Several examples of generalized number systems are constructed to compare various conditions occurring in the literature for the prime number theorem in the context of Beurling generalized primes.

Prime numbers along Rudin–Shapiro sequences

For a large class of digital functions f , we estimate the sums \sum_{n \leq x} \Lambda(n) f(n) (and \sum_{n \leq x} \mu(n) f(n) , where \Lambda denotes the von Mangoldt function (and \mu the Möbius function). We deduce from these estimates a Prime Number Theorem (and a Möbius randomness principle) for sequences of integers with digit properties including the Rudin-Shapiro sequence and some of its generalizations.

A Scale Invariant Distribution of the Prime Numbers

The irregular distribution of prime numbers amongst the integers has found multiple uses, from engineering applications of cryptography to quantum theory. The degree to which this distribution can be predicted thus has become a subject of current interest. Here, we present a computational analysis of the deviations between the actual positions of the prime numbers and their predicted positions from Riemann’s counting formula, focused on the variance function of these deviations from sequential enumerative bins. We show empirically that these deviations can be described by a class of probabilistic models known as the Tweedie exponential dispersion models that are characterized by a power law relationship between the variance and the mean, known by biologists as Taylor’s power law and by engineers as fluctuation scaling. This power law behavior of the prime number deviations is remarkable in that the same behavior has been found within the distribution of genes and single nucleotide polymorphisms (SNPs) within the human genome, the distribution of animals and plants within their habitats, as well as within many other biological and physical processes. We explain the common features of this behavior through a statistical convergence effect related to the central limit theorem that also generates 1/f noise.

Prime Numbers: A Particle in a Box and the Complex Wave Model

<p>Euler{'}s formula establishes the relationship between the trigonometric function and the exponential function. In doing so unifies two waves, a real and an imaginary one, that propagate through the Complex number set, establishing relation between integer numbers. A complex wave, if anchored by zero and by a defined integer number \textit{N}, only can assume certain oscillation modes. The first mode of oscillation corresponds always to a \textit{N} prime number and the other modes to its multiples.</p><p>\begin{center}<br />\(\psi (x)=x e^{i\left(\frac{n \pi }{N}x\right)}\)<br />\end{center}</p><p>Under the above described conditions, these waves and their admissible oscillation modes allows for primality testing of integer numbers, the deduction of a new formula $\pi(x)$ for counting prime numbers and the identification of patterns in the prime numbers distribution with computing time gains in the calculations. In this article, four theorems and one rule of factorizing are put forward with consequences for prime number signaling, counting and distribution. Furthermore, it is establish the relationship between this complex wave with a time independent semi-classical harmonic oscillator, in which the spectrum of the allowed energy levels are always only prime numbers. Thus, it is affirmative the reply to the question if the prime numbers distribution is related to the energy levels of a physical system.</p>

Extensions of Beurling's prime number theorem

Kahane provided a proof of a conjecture of Bateman and Diamond on Beurling generalized primes. The conjecture says that the L2-condition [Formula: see text] implies the PNT, i.e. π(x) ~ x/ log x. Recently, Schlage-Puchta and Vindas showed that Beurling's condition in the form of a Cesàro mean [Formula: see text] with γ > 3/2 implies the PNT as well. Both are extensions of the Beurling PNT. We now show that the condition [Formula: see text] implies the PNT and that [Formula: see text] with k ≥ 0 implies the PNT also, where [Formula: see text]These results cover the preceding forms and extend further the Beurling PNT. As part of our argument, we extend first the theorems on Chebyshev bounds with correspondingly weaker conditions. These bounds play an important role in later proofs.

A generalization of the graph theory prime-number theorem of a finite graph

In the first half, we present a generalization of the graph theory prime-number theorem due to Audrey Terras et al. In the last half, we study relations between several densities of a subset consisting of primes in a graph, and by using the first result, we compute a distribution of a subset. This is an answer for the "research problem" in the page 197 of Terras' textbook "Zeta Functions of Graphs."

Estimates of $\psi ,\theta $ for large values of $x$ without the Riemann hypothesis

The enlargement of known zero-free regions has enabled us to find better effective estimates for classical number-theoretic functions linked to the distribution of prime numbers. In particular we draw the quintessence of the method of Rosser and Schoenfeld on the upper bounds for the usual Chebyshev prime and prime power counting functions to find an upper bound function directly linked to a zero-free region.

On the Riesz Sums in Number Theory

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.

On Solving a Curious Inequality of Ramanujan

Ramanujan proved that the inequalityholds for all sufficiently large values of x. Using an explicit estimate for the error in the prime number theorem, we show unconditionally that it holds if x ≥ exp (9658). Furthermore, we solve the inequality completely on the Riemann hypothesis and show that x = 38 358 837 682 is the largest integer counterexample.

Analogues of the Prime Number Theorem and Mertens’ Theorem for Closed Orbits of the Motzkin Shift

In this paper, the prime orbit theorem and Mertens’ orbit theorem are proved for a shift dynamical system of infinite type called the Motzkin shift. Instead of using the zeta function approach to find the asymptotic formulas, different and more direct methods are used in the proof without any complicated theoretical discussion.

Fractional derivative anomalous diffusion equation modeling prime number distribution

Abstract This study suggests that the power law decay of prime number distribution can be considered a sub-diffusion process, one of typical anomalous diffusions, and could be described by the fractional derivative equation model, whose solution is the statistical density function of Mittag-Leffler distribution. It is observed that the Mittag-Leffler distribution of the fractional derivative diffusion equation agrees well with the prime number distribution and performs far better than the prime number theory. Compared with the Riemann’s method, the fractional diffusion model is less accurate but has clear physical significance and appears more stable. We also find that the Shannon entropies of the Riemann’s description and the fractional diffusion models are both very close to the original entropy of prime numbers. The proposed model appears an attractive physical description of the power law decay of prime number distribution and opens a new methodology in this field.

Riemann equation for prime number diffusion.

This study makes the first attempt to propose the Riemann diffusion equation to describe in a manner of partial differential equation and interpret in physics of diffusion the classical Riemann method for prime number distribution. The analytical solution of this equation is the well-known Riemann representation. The diffusion coefficient is dependent on natural number, a kind of position-dependent diffusivity diffusion. We find that the diffusion coefficient of the Riemann diffusion equation is nearly a straight line having a slope 0.99734 in the double-logarithmic axis. Consequently, an approximate solution of the Riemann diffusion equation is obtained, which agrees well with the Riemann representation in predicting the prime number distribution. Moreover, we interpret the scale-free property of prime number distribution via a power law function with 1.0169 the scale-free exponent in respect to logarithmic transform of the natural number, and then the fractal characteristic of prime number distribution is disclosed.