Entropy, Periodicity and the Probability of Primality.

<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>

Fractals and discrete dynamics associated to prime numbers

The distribution of twin prime numbers is discussed. The research method of corresponding prime number distribution is proposed. The distribution of prime numbers corresponding to integers and composite numbers is discussed. Through the corresponding prime distribution rate of integers and composite numbers, it is found that the corresponding prime distribution rate of composite numbers approaches the corresponding prime distribution rate of integers. The distribution principle of corresponding prime number of composite number is proved. The twin prime distribution theorem is obtained. The number of twin prime numbers is thus obtained. It provides a practical way to study the conjecture of twin prime numbers.

This paper deal with the development of prime and composite numbers and their modern applications to mathematical and physical sciences. It contains the distribution of prime numbers, prime number theorems, Euler’s and Riemann’s zeta functions and their remarkable link with prime numbers and the celebrated unsolved Riemann Hypothesis (RH). Special attention is given to the discovery of the Fermat and the Mersenne prime numbers, and numerous modern computational results in support of the RH. Proofs of different versions of prime number theorems discovered by many greatest mathematicians of the world are mentioned. Mention is also made of one of the remarkable aspects of the distribution of prime numbers and their tendency to exhibit local irregularity and global regularity. This naturally leads to the stochastic distribution of prime numbers and the Gauss-Cramer probabilistic model to determine the stochastic prime number theorems in short intervals. It is found that the Gauss-Cramer model is consistent with the RH and the twin prime conjecture. Included are many unsolved problems and conjectures that put students, teachers, and mathematical scientists and professionals at the forefront of current advanced study and research in analytical and computational number theory.

This research explores the distribution of prime numbers, which are a fundamental topic in number theory. The study originated from the author's fascination with mathematics and the desire to discover something novel. The research proposes that the distribution of prime numbers follows a regular pattern starting from the number 2. The author suggests that prime numbers can be obtained by dividing certain even numbers that have four factors by the number 2, resulting in prime numbers in sequential order. This hypothesis was tested and confirmed through the practical application of the proposed mathematical formula. Additionally, the study found that even numbers greater than or equal to 8, with six or more factors, produce complex numbers. Thus, this research provides two main contributions: firstly, a mathematical formula for the distribution of prime numbers, and secondly, a formula for the distribution of complex numbers. These findings have potential applications in various mathematical fields, including cryptography and problem-solving in number theory.

Nowadays, many researchers are puzzled by the problems about prime numbers, not only due to its complex and unknown characteristics, but also because it is difficult to construct a model to describe the relaxation distribution of prime numbers in the set of all natural numbers. To the best knowledge of authors, although some phenomenological methods, such as the Riemann's Theory, can describe the distribution of the prime numbers accurately, the law for the distribution of the prime numbers is not clear and the governing model has not founded until now. In the present study, for the first time, we employ anomalous relaxation model with the fractional derivative to obtain the distribution of prime numbers. The comparison with Prime Number Theory and Riemann's Theory show that the new model agrees well with the data of prime numbers.

This research explores the distribution of prime numbers, which are a fundamental topic in number theory. The study originated from the author's fascination with mathematics and the desire to discover something novel. The research proposes that the distribution of prime numbers follows a regular pattern starting from the number 2. The author suggests that prime numbers can be obtained by dividing certain even numbers that have four factors by the number 2, resulting in prime numbers in sequential order. This hypothesis was tested and confirmed through the practical application of the proposed mathematical formula. Additionally, the study found that even numbers greater than or equal to 8, with six or more factors, produce complex numbers. Thus, this research provides two main contributions: firstly, a mathematical formula for the distribution of prime numbers, and secondly, a formula for the distribution of complex numbers. These findings have potential applications in various mathematical fields, including cryptography and problem-solving in number theory.

There is a similarity between the distribution of prime numbers and the pattern of earthquake occurrence. Earthquakes occur in a discrete manner in time and space. When viewed as a whole, however, we find some laws, such as Gutenberg-Richter law, that govern the entire earthquakes that seem to be individually independent. A similar phenomenon can be observed also in the world of number. The most basic example is the distribution of the prime numbers in integers. We consider a correspondence between earthquakes and prime numbers. We parameterize occurrence time of earthquakes as the prime numbers and magnitude of earthquakes as the interval of prime numbers. Then we obtain a relationship similar to Gutenberg-Richter law. We call the model obtained by this correspondence as “arithmetic seismic activity model”. If we can parameterize earthquakes using prime numbers, knowledge that has been cultivated in the number theory can be used for understanding of earthquakes. The distribution of prime numbers is related to the distribution of zeros of Riemann zeta function. Researches are in progress to understand the zeros of the Riemann zeta function as an eigenvalue problem of quantum dynamical system. Earthquake may be modeled as a phenomenon corresponding to a change in the energy level of a quantum dynamical system associated with prime numbers.

A Boolean function on $N$ variables is called \emph{evasive} if its decision-tree complexity is $N$. A sequence $B_n$ of Boolean functions is \emph{eventually evasive} if $B_n$ is evasive for all sufficiently large $n$. We confirm the eventual evasiveness of several classes of monotone graph properties under widely accepted number theoretic hypotheses. In particular we show that Chowla's conjecture on Dirichlet primes implies that (a) for any graph $H$, ``forbidden subgraph $H$'' is eventually evasive and (b) all nontrivial monotone properties of graphs with $\le n^{3/2-\epsilon}$ edges are eventually evasive. ($n$ is the number of vertices.) While Chowla's conjecture is not known to follow from the Extended Riemann Hypothesis (ERH, the Riemann Hypothesis for Dirichlet's $L$ functions), we show (b) with the bound $O(n^{5/4-\epsilon})$ under ERH. We also prove unconditional results: (a$'$) for any graph $H$, the query complexity of ``forbidden subgraph $H$'' is $\binom{n}{2} - O(1)$; (b$'$) for some constant $c>0$, all nontrivial monotone properties of graphs with $\le cn\log n+O(1)$ edges are eventually evasive. Even these weaker, unconditional results rely on deep results from number theory such as Vinogradov's theorem on the Goldbach conjecture. Our technical contribution consists in connecting the topological framework of Kahn, Saks, and Sturtevant (1984), as further developed by Chakrabarti, Khot, and Shi (2002), with a deeper analysis of the orbital structure of permutation groups and their connection to the distribution of prime numbers. Our unconditional results include stronger versions and generalizations of some result of Chakrabarti et al.

After the first world war, Cramer began studying the distribution of prime numbers, guided by Riesz and Mittag-Leffler. His works then, and later in the mid-thirties, have had a profound influence on the way mathematicians think about the distribution of prime numbers. In this article, we shall focus on how Cramer's ideas have directed and motivated research ever since.

Negative correlations in the distribution of prime numbers are found to display a scale invariance. This occurs in conjunction with a nonstationary behavior. We compare the prime number series to a type of fractional Brownian motion which incorporates both the scale invariance and the nonstationary behavior. Interesting discrepancies remain. The scale invariance also appears to imply the Riemann hypothesis and we study the use of the former as a test of the latter.

In this paper, the distribution of prime numbers is expressed based on proving the Riemann hypothesis. The relationship between three numbers, three, six, and nine, and the modality to the distribution of prime numbers, is one of the results of Riemann's zeta function. Prime numbers are classified into six groups of single-digit numbers. There are no prime numbers in groups of three, six, and nine. The groups are made based on the sum of the internal digits. And for each set, there is an angle in the complex plane. The distance between the prime numbers in each group has a regular pattern. This pattern is a multiple of the numbers three, six, and nine. According to Euler's number, for an angle of 60 degrees, the real part of the cosine is 1/2. Accordingly, all prime numbers are related to angles greater than 60 degrees to 90 degrees. As a result, based on the relationship between the golden spiral and the complex conjugate of the zeta function, the function in The 1/2 point becomes zero based on the prime numbers.

This paper outlines further properties concerning the fractional derivative of the Riemann ζ function. The functional equation, computed by the introduction of the Grünwald–Letnikov fractional derivative, is rewritten in a simplified form that reduces the computational cost. Additionally, a quasisymmetric form of the aforementioned functional equation is derived (symmetric up to one complex multiplicative constant). The second part of the paper examines the link with the distribution of prime numbers. The Dirichlet η function suggests the introduction of a complex strip as a fractional counterpart of the critical strip. Analytic properties are shown, particularly that a Dirichlet series can be linked with this strip and expressed as a sum of the fractional derivatives of ζ. Finally, Theorem 4.3 links the fractional derivative of ζ with the distribution of prime numbers in the left half-plane.

<p>The following paper deals with the distribution of prime numbers, the twin prime numbers and the Goldbach conjecture. Starting from the simple assertion that prime numbers are never even, a rule for the distribution of primes is arrived at. Following the same approach, the twin prime conjecture and the Goldbach conjecture are found to be true.</p>

This paper is devoted to irregularities in the distribution of prime numbers. We describe the development of this theory and the relation to Maier’s matrix method.