Prime Number Theorem Research Articles

Elementary components of electroencephalography signals viewed as prime numbers

The disabling aspect of epilepsy disease is that it seems to be erratic and random in nature. Electroencephalogram (EEG) serves as an essential non-invasive tool to diagnose and manage epilepsy, allowing the physiological manifestations of irregular cortical excitability to be demonstrated. Certain prior EEG features related to seizure onset may facilitate seizure pattern predictions through mathematical models. A better understanding of these patterns can positively improve epilepsy management and, in turn, improve the quality of life of epilepsy patients. Thus, the goal of the current paper is to show that elementary EEG signals during a seizure can be perceived as prime numbers. First, the recorded EEG signals are written as a product of their elementary components through the Krohn – Rhodes decomposition technique. Following this, every elementary component of EEG signal is expressed in terms of a summation of their simpler parts via the Jordan – Chevalley decomposition process. Conversely, some prime numbers are decomposed similar to Jordan – Chevalley decomposition of elementary EEG signals and presented as Pseudo – Goldbach Theorem. Finally, the results demonstrate substantial evidence that the EEG signals follow a pattern similar to that of the distribution of prime numbers among positive integers.

Fractional Parts of Non-Integer Powers of Primes. II

Abstract We continue to study the distribution of prime numbers p, satisfying the condition $\{p^{\alpha} \} \in I \subset [0; 1)$, in arithmetic progressions. In this paper, we prove an analogue of Bombieri–Vinogradov theorem for 0 < α < 1/9 with the level of distribution $\theta = 2/5 - (3/5) \alpha$, which improves the previous result corresponding to $\theta \leqslant 1/3$.

Prime numbers demystified

The paper is the ultimate prime numbers algorithm that gets rid of the unneccessary mystery about prime numbers. All the numerous arithmetic series patterns observed between various prime numbers are clearly explained with an elegant "pattern of remainders". With this algorithm we prove that odd numbers too can make an Ulam spiral contrary to current ""proofs". At the end of the paper this author proves the relationship between a simple arithmetic series pattern and the Riehmann's prime numbers distribution equation. This paper would be important for encryption too. As an example, prime integer 1979 is expressed as 1.2.4.5.10.3.7.3.1.7.26.18.11.1. This makes even smaller primes useful for encryption as well.

Combinatorial Models of the Distribution of Prime Numbers

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.

A new elementary proof of the Prime Number Theorem

Let Ω ( n ) denote the number of prime factors of n. We show that for any bounded f : N → C one has 1 N ∑ n = 1 N f ( Ω ( n ) + 1 ) = 1 N ∑ n = 1 N f ( Ω ( n ) ) + o N → ∞ ( 1 ) . This yields a new elementary proof of the Prime Number Theorem.

Elliptic curves with Galois-stable cyclic subgroups of order 4

Infinitely many elliptic curves over $$\mathbf{Q }$$ have a Galois-stable cyclic subgroup of order 4. Such subgroups come in pairs, which intersect in their subgroups of order 2. Let $$N_j(X)$$ denote the number of elliptic curves over $$\mathbf{Q }$$ with at least j pairs of Galois-stable cyclic subgroups of order 4, and height at most X. In this article we show that $$N_1(X) = c_{1,1}X^{1/3}+c_{1,2}X^{1/6}+O(X^{0.105})$$ . We also show, as $$X\rightarrow \infty $$ , that $$N_2(X)=c_{2,1}X^{1/6}+o(X^{1/12})$$ , the precise nature of the error term being related to the prime number theorem and the zeros of the Riemann zeta-function in the critical strip. Here, $$c_{1,1}= 0.95740\ldots $$ , $$c_{1,2}=- 0.87125\ldots $$ , and $$c_{2,1}= 0.035515\ldots $$ are calculable constants. Lastly, we show no elliptic curve over Q has more than 2 pairs of Galois-stable cyclic subgroups of order 4.

Prime Numbers Calculation Formulas

The application of prime numbers in modern science, especially in computer science, is very wide. Since prime numbers can only divisible by 1 and themselves, they are not factored any further like whole numbers. The problem to calculate all prime numbers using a formula posed for long periods. Though different formulas to calculate prime numbers were developed by Euler, Fermat, Mersenne and others, the formulas work for limited natural numbers and calculate limited prime numbers. JEL classification numbers: C02, C63, C69 Keywords: Prime numbers, Prime numbers formula, Prime number distribution, Prime number calculation.

A Survey on Newmann’s proof of the prime number theorem

The prime number theorem is one of the most important theorems in number theory, which tells the asymptotic behavior of positive prime numbers. It was first proved independently by Jacques Hadamard and Charles-Jean de la Valĺee Poussin in 1896, by using the Riemann zeta function as a tool. We summarize Newmann’s proof in this survey.

Sign changes in the prime number theorem

Let V(T) denote the number of sign changes in $$\psi (x) - x$$ for $$x\in [1, T]$$ . We show that $$\liminf _{T\rightarrow \infty } V(T)/\log T\ge \gamma _{1}/\pi + 1.867\cdot 10^{-30}$$ , where $$\gamma _{1} = 14.13\ldots $$ denotes the ordinate of the lowest-lying non-trivial zero of the Riemann zeta-function. This improves on a long-standing result by Kaczorowski.

Prime number theorem for regular Toeplitz subshifts

AbstractWe prove that neither a prime nor an l-almost prime number theorem holds in the class of regular Toeplitz subshifts. But when a quantitative strengthening of the regularity with respect to the periodic structure involving Euler’s totient function is assumed, then the two theorems hold.

A study of prime numbers distribution based on support vector domain description

Prime numbers are fundamental to the most common type of asymmetric encryption used today, namely Diffie-Hellman key exchange, Rivest-Shamir-Adleman cryptosystems, ElGamal Cryptosystem, Elliptic curve cryptosystems, etc. The strength of primes lies in the fact that they are the ‘building blocks’ of mathematics and in addition there is infinity of them with no apparent regularity in their distribution. Given these characteristics, they were used to build efficient one way functions used in public key cryptography. This work aims at exploring sets of prime numbers with the objective of finding patterns in their structure. For this purpose an efficient supervised learning model called Support Vector Domain Description was used as an explorer. The experimental results have shown a good generalization capability of this model to describe new prime numbers.

The Prime Number Theorem as a Capstone in a Complex Analysis Course

We present a detailed proof of the prime number theorem suitable for a typical undergraduate- or graduate-level complex analysis course. Our presentation is particularly useful for any instructor who seeks to use the prime number theorem for a series of capstone lectures, a scaffold for a series of guided exercises, or as a framework for an inquiry-based course. We require almost no knowledge of number theory, for our aim is to make a complete proof of the prime number theorem widely accessible to complex analysis instructors and their students. In particular, we highlight the potential pitfalls and subtleties that may catch the instructor unawares when using more terse sources.

Riemann Hypothesis, Catholic Information and Potential of Events with New Techniques for Financial and Other Applications

In this research we are going to define two new concepts: a) “The Potential of Events” (EP) and b) “The Catholic Information” (CI). The term CI derives from the ancient Greek language and declares all the Catholic (general) Logical Propositions () which will true for every element of a set A. We will study the Riemann Hypothesis in two stages: a) By using the EP we will prove that the distribution of events e (even) and o (odd) of Square Free Numbers (SFN) on the axis Ax(N) of naturals is Heads-Tails (H-T) type. b) By using the CI we will explain the way that the distribution of prime numbers can be correlated with the non-trivial zeros of the function ζ(s) of Riemann. The Introduction and the Chapter 2 are necessary for understanding the solution. In the Chapter 3 we will present a simple method of forecasting in many very useful applications (e.g. financial, technological, medical, social, etc) developing a generalization of this new, proven here, theory which we finally apply to the solution of RH. The following Introduction as well the Results with the Discussion at the end shed light about the possibility of the proof of all the above. The article consists of 9 chapters that are numbered by 1, 2, …, 9.

Improved Riemann Ladder Function

In Riemann’s prime distribution formula, there is a key Riemann ladder function. This is the basis for Riemann to study the distribution of prime numbers. But the calculation of Riemann’s ladder function is very complicated. According to the definition of Riemannian ladder function, we greatly improve the Riemannian ladder function, obtain new ladder function and improved Riemannian prime distribution formula, and prove the improved Riemannian prime distribution formula. We use the improved Riemannian prime distribution formula and merdens theorem to obtain a strong prime theorem.

On the distribution of twin primes

We introduce a sieve for counting twin primes up to a given range. Our method depends on a parameter ${\lambda}_x$ and the estimation of the number of twin primes obtained as a result, is called a fundamental structure of the distribution of twin primes. Combining the latter with an asymptotic bound of ${\lambda}_x$, establishes venues, conducive to a discovery of a partial result that can be considered as a suitable variant of the prime number theorem. Furthermore, we obtain an asymptotic bound of the number of twin primes less than $x$.

The Prime state and its quantum relatives

The Prime state of n qubits, |Pn⟩, is defined as the uniform superposition of all the computational-basis states corresponding to prime numbers smaller than 2n. This state encodes, quantum mechanically, arithmetic properties of the primes. We first show that the Quantum Fourier Transform of the Prime state provides a direct access to Chebyshev-like biases in the distribution of prime numbers. We next study the entanglement entropy of |Pn⟩ up to n=30 qubits, and find a relation between its scaling and the Shannon entropy of the density of square-free integers. This relation also holds when the Prime state is constructed using a qudit basis, showing that this property is intrinsic to the distribution of primes. The same feature is found when considering states built from the superposition of primes in arithmetic progressions. Finally, we explore the properties of other number-theoretical quantum states, such as those defined from odd composite numbers, square-free integers and starry primes. For this study, we have developed an open-source library that diagonalizes matrices using floats of arbitrary precision.

Error term of the mean value theorem for binary Egyptian fractions

Abstract In this article, the error term of the mean value theorem for binary Egyptian fractions is studied. An error term of prime number theorem type is obtained unconditionally. Under Riemann hypothesis, a power saving can be obtained. The mean value in short interval is also considered.

The error term in the prime number theorem

We make explicit a theorem of Pintz concerning the error term in the prime number theorem. This gives an improved version of the prime number theorem with error term roughly square-root of that which was previously known. We apply this to a long-standing problem concerning an inequality studied by Ramanujan.

Distribution of adjacent prime numbers in the Ulam spiral

Purpose The distribution of natural numbers in the Ulam spiral manifests a series of unexpected regularities of the elusive prime numbers. Their sequencing remains a topic of research interest, with many ramifications in different branches of Mathematics, especially in number theory and the prime factorisation problem. Accordingly, the focus of the research is on the most known and widespread asymmetric cryptographic system, that is, the RSA encryption. Design/methodology/approach This paper presents the presence of one, two, three or four adjacencies for the diverse primes that appear in a spiral, considering the Hardy–Littlewood twin prime conjecture and the constellations of primes. Findings Through equations, the calculation formulas of primes inside a spiral that have one to four primes in their adjacent places is offered, with approximate expressions that facilitate the operations, showing that the adjacencies decrease very rapidly as the spiral progresses, although without disappearing. Originality/value A generalisation to cases in which the distances to the prime values change in an ascending way in accordance with the step of the Ulam spiral is offered.

Distribution of prime numbers and the discrete spectrum of the Laplace operator

We obtain a class of explicit formulae each of which gives an expression for the remainder term in the asymptotic equation for the Chebyshev function in terms of the spectrum of the Laplace operator on the fundamental domain of the modular group.