The distribution of prime numbers

A function to calculate all relative prime numbers up to the product of the first n primes

We prove an isomorphism between the finite domain from 1 up to the product of the first n primes and the new defined set of prime modular numbers. This definition provides some insights about relative prime numbers. We provide an inverse function from the prime modular numbers into this finite domain. With this function we can calculate all numbers from 1 up to the product of the first n primes that are not divisible by the first n primes. This function provides a non sequential way for the calculation of prime numbers.

Characterization of the Distribution of Twin Primes

We adopt an empirical approach to the characterization of the distribution of twin primes within the set of primes, rather than in the set of all natural numbers. The occurrences of twin primes in any finite sequence of primes are like fixed probability random events. As the sequence of primes grows, the probability decreases as the reciprocal of the count of primes to that point. The manner of the decrease is consistent with the Hardy--Littlewood Conjecture, the Prime Number Theorem, and the Twin Prime Conjecture. Furthermore, our probabilistic model, is simply parameterized. We discuss a sim…

Exceptional Prime Number Twins, Triplets and Multiplets

A classification of twin primes implies special twin primes. When applied to triplets, it yields exceptional prime number triplets. These generalize yielding exceptional prime number multiplets.

关于满足 p+2 至多有两个素因子的素数 p 的 αp²+β 模一分布

On the distribution of $αp^2+β$ modulo one for primes $p$ such that $p+2$ has no more two prime divisors

本文针对 α 无理时 αp²+β 模 1 的分布问题，考察 p+2 为殆素数的素数子序列，证明该子序列保持与经典素数序列相似的分布性质。

Remarkable and Reversible Prime Number Patterns

Prime number multiplet classifications and patterns are extended to negative integers. The extension from prime numbers to single prime powers is also studied. Prime number septets at equal distance are given. It is also shown that each class of generalized twin primes of the classification contains a positive fraction of all prime pairs.

Implications of a New Characterisation of the Distribution of Twin Primes

We bring to bear an empirical model of the distribution of twin primes and produce two distinct results. The first is that we can make a quantitative probabilistic prediction of the occurrence of gaps in the sequence of twins within the primes. The second is that the ``high jumper'' i.e., the separation with greatest likelihood (in terms of primes) is always expected to be zero.

Prime and zero distributions for meromorphic Euler products

The aim of the present paper is to study the relations between the prime distribution and the zero distribution for generalized zeta functions which are expressed by Euler products and is analytically continued as meromorphic functions of finite order. In this paper, we give an inequality between the order of the zeta function as a meromorphic function and the growth of the multiplicity in the prime distribution.

Nonuniform Distributions of Residues of Prime Sequences in Prime Moduli

For positive integers $q$, Dirichlet's theorem states that there are infinitely many primes in each reduced residue class modulo $q$. A stronger form of the theorem states that the primes are equidistributed among the $φ(q)$ reduced residue classes modulo $q$. This paper considers patterns of sequences of consecutive primes $(p_n, p_{n+1}, \ldots, p_{n+k})$ modulo $q$. Numerical evidence suggests a preference for certain prime patterns. For example, computed frequencies of the pattern $(a,a)$ modulo $q$ up to $x$ are much less than the expected frequency $π(x)/φ(q)^2$. We begin to rigorously c…

Asymptotics of the number of primes and sums of functions of primes in a subset of natural series

The paper solves the problems of determining the asymptotics of the number of primes and the sums of functions of primes in a subset of the natural series that satisfies the conditions that the asymptotic density of the number of primes in this subset is constant and not equal to zero.

A heuristic study of the distribution of primes in short and not-so-short intervals

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 $h \gg \log{N}$ but $h \ll N$, providing support to the Montgomery and Soundararajan conjecture on the variance of the prime distribution at this scale.

左素截断的分布

Distributions of left prime truncations

研究了整数和多项式左素截断的分布特性，分析了其在所有ℓ位数或多项式中的比例、方差及最大比例。

Egyptian Fractions and Prime Power Divisors

From varying Egyptian fraction equations we obtain generalizations of primary pseudoperfect numbers and Giuga numbers which we call prime power psuedoperfect numbers and prime power Giuga numbers respectively. We show that a sequence of Amarnath Murthy in the OEIS is a subsequence of the sequence of prime power psuedoperfect numbers. Prime factorization conditions sufficient to imply a number is a prime power pseudoperfect number or a prime power Giuga number are given. The conditions on prime factorizations naturally give rise to a generalization of Fermat primes which we call extended Fermat…

A heuristic study of prime number distribution

This work consists of a heuristic study on the distribution of prime numbers in short intervals. We have modelled the occurrence of prime numbers such intervals as a counting experiment. As a result, we have provided an experimental validation and an extension of the Montgomery and Soundararajan conjecture. This is a reduced version of my bachelor`s thesis presented at the University of Valencia.

On the distribution of Atkin and Elkies primes for reductions of elliptic curves on average

For an elliptic curve E/Q without complex multiplication we study the distribution of Atkin and Elkies primes l, on average, over all good reductions of E modulo primes p. We show that, under the Generalised Riemann Hypothesis, for almost all primes p there are enough small Elkies primes l to ensure that the Schoof-Elkies-Atkin point-counting algorithm runs in (log p)^(4+o(1)) expected time.

Distribution of twisted Kloosterman sums modulo prime powers

In this note we study Kloosterman sums twisted by a multiplicative characters modulo a prime power. We show, by an elementary calculation, that these sums become equidistributed on the real line with respect to a suitable measure.

Large gaps between consecutive prime numbers containing perfect $k$-th powers of prime numbers

Let $k\geq 2$ be a fixed natural number. We establish the existence of infinitely many pairs of consecutive primes $p_n$, $p_{n+1}$ satisfying $$ p_{n+1}-p_n\geq c\:\frac{\log p_n\: \log_2 p_n\: \log_4 p_n}{\log_3 p_n}\:,$$ with $c$ being a fixed positive constant, for which the interval $(p_n, p_{n+1})$ contains the $k$-th power of a prime number.

On the distribution of sparse sequences in prime fields and applications

In the present paper we investigate distributional properties of sparse sequences modulo almost all prime numbers. We obtain new results for a wide class of sparse sequences which in particular find applications on additive problems and the discrete Littlewood problem related to lower bound estimates of the $L_1$-norm of trigonometric sums.

Note on the Prime Number Theorem

We survey the classical results on the prime number theorem

Statistical Bias in the Distribution of Prime Pairs and Isolated Primes

Computer experiments reveal that twin primes tend to center on nonsquarefree multiples of 6 more often than on squarefree multiples of 6 compared to what should be expected from the ratio of the number of nonsquarefree multiples of 6 to the number of squarefree multiples of 6 equal $π^2/3-1$, or ca 2.290. For multiples of 6 surrounded by twin primes, this ratio is 2.427, a relative difference of ca $6.0\%$ measured against the expected value. A deviation from the expected value of this ratio, ca $1.9\%$, exists also for isolated primes. This shows that the distribution of primes is biased towa…

On the nonasymptotic prime number distribution

The objective of this paper is to introduce an approach to the study of the nonasymptotic distribution of prime numbers. The natural numbers are represented by theorem 1 in the matrix form ^2N. The first column of the infinite matrix ^2N starts with the unit and contains all composite numbers in ascending order.The infinite rows of this matrix except for the first elements contain prime numbers only, which are determined by an uniform recurrence law. At least one of the elements of the twin pairs of prime numbers is an element of the second column of the matrix ^2N (theorem 3). The basic infor…

Less Regular Exceptional and Repeating Prime Number Multiplets

New exceptional (i.e. non-repeating) prime number multiplets are given and formulated in terms of arithmetic progressions, along with laws governing them. Accompanying repeating prime number multiplets are pointed out. Prime number multiplets with less regular distances are studied.

Medium-sized values for the Prime Number Theorem for primes in arithmetic progression

We give two improved explicit versions of the prime number theorem for primes in arithmetic progression: the first isolating the contribution of the Siegel zero and the second completely explicit, where the improvement is for medium-sized values. This will give an improved explicit Bombieri-Vinogradov like result for non-exceptional moduli.

The Eratosthenes Progression p_{k+1} = π^{-1}(p_k), k=0,1,2,..., p_0 \in N \setminus P Determines An Inner Prime Number Distribution Law

Formulas of $π(x)$-fine structure are presented.

Supersingular distribution on average for congruence classes of primes

We demonstrate the existence of a congruence class bias in the distribution of supersingular primes on average for elliptic curves over $\Q$. For example, we show that on average there are twice as many supersingular primes congruent to 2 mod 3 as there are congruent to 1 mod 3. Our result is obtained using the averaging approach of Fouvry-Murty along with ideas of David-Pappalardi.

On Proving of Diophantine Inequalities with Prime Numbers by Evaluations of the Difference between Consecutive Primes

Using as the working hypothesis of an evaluation of the difference between primes $p_{n+1} - p_n = O(\sqrt{p_n})$ we represent in detail the proofs of Legendre's and Oppermann's conjectures.

Distribution of primes represented by polynomials and Multiple Dedekind zeta functions

n this paper, we state several conjectures regarding distribution of primes and of pairs of primes represented by irreducible homogeneous polynomial in two variables $f(a,b)$. We formulate conjectures with respect to the slope $t=b/a$ for any irreducible polynomial $f$. Here, we formulate a conjecture for all irreducible polynomials. We also consider conjectures for distribution of pairs of primes. It show unexpected relation to multiple Dedekind zeta function - at $s=2$ for one prime and at $(s_1,s_2)=(2,2)$ for pairs of primes. We tested the conjecture for pairs of primes for several quadrat…

The distribution of lattice points with relatively r-prime

The distribution of lattice points with relatively $r$-prime is related to problems in the Number Theory such as the Extended Lindelöf Hypothesis and the Gauss Circle Problem. It is known that Sittinger's result is improved on the assumption of the Extended Lindelöf Hypothesis. In this paper, we improve Sittinger's result without assuming the Extended Lindelöf hypothesis.

The distribution of prime numbers and tuples in the natural numbers

We studied two probabilistic models of the distribution of primes in the natural number [1].The paper considers the third probabilistic model of the distribution of primes in the natural number. The author proved that the results obtained by considering three probabilistic models are similar. It is shown that the accuracy of these probabilistic models exceeds the Riemann hypothesis with a certain probability. The author constructed probabilistic models for the analysis of the average distance between neighboring prime numbers and k-tuples and found probability distributions of these random var…

Distribution of integral Fourier Coefficients of a Modular Form of Half Integral Weight Modulo Primes

Recently, Bruinier and Ono classified cusp forms $f(z) := \sum_{n=0}^{\infty} a_f(n)q ^n \in S_{λ+1/2}(Γ_0(N),χ)\cap \mathbb{Z}[[q]]$ that does not satisfy a certain distribution property for modulo odd primes $p$. In this paper, using Rankin-Cohen Bracket, we extend this result to modular forms of half integral weight for primes $p \geq 5$. As applications of our main theorem we derive distribution properties, for modulo primes $p\geq5$, of traces of singular moduli and Hurwitz class number. We also study an analogue of Newman's conjecture for overpartitions.

On the distribution of $αp^2$ modulo one with prime $p$ of a special form

Let $\mathcal{P}_r$ denote an almost-prime with at most $r$ prime factors, counted according to multiplicity. In this paper, it is proved that for $α\in\mathbb{R}\backslash\mathbb{Q},\,β\in\mathbb{R}$ and $0<θ<10/1561$, there exist infinitely many primes $p$, such that $\|αp^2+β\|<p^{-θ}$ and $p+2=\mathcal{P}_4$, which constitutes an improvement upon the previous result.

高阶素数序列的性质

Properties of Higher-Order Prime Number Sequences

本文研究由素数高阶子序列交错和生成的素数序列的性质，并提出一种通过在实数轴上系统选择与消除素数下标来构造这些序列的新筛法。

On the Difference between Consecutive Primes and Estimates of the Number of Primes in the Interval $(n, 2n)$

Using evaluations of the difference between consecutive primes we develop another way of estimating of the number of primes in the interval $(n, 2n)$. We also discuss the ultra Cramer conjecture, $p_{n+1} - p_n = O(log^{1+ε}p_n)$ where $ε> 0$, in the context of the results we have obtained in our paper.

关于实二次域中素理想分布的一个注记

A note on the distribution of prime ideals in real quadratic fields

本文总结了虚二次域素理想分布的研究，并基于相同方法建立了实二次域的类似结果。

On the differences between consecutive prime numbers, I

We show by an inclusion-exclusion argument that the prime $k$-tuple conjecture of Hardy and Littlewood provides an asymptotic formula for the number of consecutive prime numbers which are a specified distance apart. This refines one aspect of a theorem of Gallagher that the prime $k$-tuple conjecture implies that the prime numbers are distributed in a Poisson distribution around their average spacing.

A remark on prime (non)congruent numbers

This paper discusses prime numbers that are (resp. are not) congruent numbers. Particularly the only case not fully covered by earlier results, namely primes of the form $p=8k+1$, receives attention.

Novák-Carmichael numbers and shifted primes without large prime factors

We prove some new lower bounds for the counting function $\mathcal N_{\mathcal C}(x)$ of the set of Novák-Carmichael numbers. Our estimates depend on the bounds for the number of shifted primes without large prime factors. In particular, we prove that $\mathcal N_{\mathcal C}(x) \gg x^{0.7039-o(1)}$ unconditionally and that $\mathcal N_{\mathcal C}(x) \gg xe^{-(7+o(1))(\log x)\frac{\log\log\log x}{\log\log x}}$, under some reasonable hypothesis.

On the distribution of the digits of quotients of integers and primes

We investigate the distribution of the digits of quotients of randomly chosen positive integers taken from the interval $[1,T]$, improving the previously known error term for the counting function as $T\to+\infty$. We also resolve some natural variants of the problem concerning points with prime coordinates and points that are visible from the origin.

关于一个素数和一个与任何整数互质且至多有两个素因子的平方数之和

On the sum of a prime and a square-free number co-prime to any integer with at most two prime factors

研究证明大于2的自然数可表示为素数与特定平方数之和，并明确给出了渐进行为结果。

Residue races of the number of prime divisors function

We investigate the distribution of the function $ω(n)$, the number of distinct prime divisors of $n$, in residue classes modulo $q$ for natural numbers $q$ greater than 2. In particular we ask `prime number races' style questions, as suggested by Coons and Dahmen in their paper `On the residue class distribution of the number of prime divisors of an integer'.

Squarefree smooth numbers and Euclidean prime generators

We show that for each prime p > 7, every residue mod p can be represented by a squarefree number with largest prime factor at most p. We give two applications to recursive prime generators akin to the one Euclid used to prove the infinitude of primes.

Asymptotic density of Motzkin numbers modulo small primes

We establish the asymptotic density of the Motzkin numbers modulo small primes.

An analogue of the prime number theorem for finite fields

We prove an analogue of the prime number theorem for finite fields.

Sequence elimination function and the formulas of prime numbers

We will derive a function that eliminates any sequence of equidistant numbers from the integer numbers, then we will derive its inverse. Then we will use the Sequence elimination function to eliminate the multiples of the prime numbers from the integer numbers to produce formulas that generate all prime numbers in some interval.

Distribution of prime ideals of higher residue degree across ideal classes in the class groups

In this article we investigate the distribution of prime ideals of residue degree bigger than one across the ideal classes in the class group of a number field $L$. A criterion for the class group of $L$ being generated by the classes of prime ideals of residue degree $f>1$ is provided. Further, some consequences of this study on the solvability of norm equations for $L/\mathbb{Q}$ and on the problem of finding annihilators for relative extensions are discussed.

Unexpected biases in the distribution of consecutive primes

While the sequence of primes is very well distributed in the reduced residue classes (mod $q$), the distribution of pairs of consecutive primes among the permissible $ϕ(q)^2$ pairs of reduced residue classes (mod $q$) is surprisingly erratic. This paper proposes a conjectural explanation for this phenomenon, based on the Hardy-Littlewood conjectures. The conjectures are then compared to numerical data, and the observed fit is very good.

Prime Counting Function Identity

In this paper, a new formula for π^(2)(N) is formulated, it is a function that counts the number of semi-primes not exceeding a given number N. A semi-prime is a natural number that is the product of precisely two prime numbers, the two primes in the product may equal each other. Semi-prime numbers are also a case of almost primes. Since a formula for this is already known, a new identity that uses the prime counting function is created by equating the two functions.

具有指定素因子数量的 Carmichael 数

Carmichael Numbers with a Specified Number of Prime Factors

本文证明，对任意充分大的整数 $R$，都存在恰好含有 $R$ 个素因子的 Carmichael 数。

On the Sum and Product of Distinct Prime Factors of an Odd Perfect Number

We present lower bounds on the sum and product of the distinct prime factors of an odd perfect number, which provide a lower bound on the size of the odd perfect number as a function of the number of its distinct prime factors.

Beurling numbers whose number of prime factors lies in a specified residue class

We find asymptotics for $S_{K,c}(x)$, the number of positive integers below $x$ whose number of prime factors is $c \; \mathrm{mod}\; K$. We study this question in the context of Beurling integers.

On the Prime Numbers in the Interval [4n,5n]

Is it true that for all $n\geq k\geq 2$ there exists a prime number between $kn$ and $(k+1)n$? In this paper we show that there is always a prime number between $4n$ and $5n$ for all $n>2$. We also show there are at least seven prime numbers between $n$ and $5n$ for all $n>5$.

素数无限性的另一种拓扑学证明

Another Topological Proof of the Infinitude of Prime Numbers

本文通过引入一种新拓扑，为素数无限性提供了新的拓扑学证明，并刻画了素数任意非空子集的无限性。

Poisson distribution of a prime counting function corresponding to elliptic curves

Let $E$ be an elliptic curve defined over rational field $\mathbb{Q}$ and $N$ be a positive integer. Now, $M_E(N)$ denotes the number of primes $p$, such that the group $E_p(\mathbb{F}_p)$ is of order $N$. We show that $M_E(N)$ follows Poisson distribution when an average is taken over a large class of curves.

Sets of Completely Decomposed Primes in Extensions of Number Fields

We introduce the notion of saturated sets of primes of an algebraic number field and prove an analogue of Riemann's existence theorem for the decomposition groups of infinite stably saturated sets of primes.

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.