Related papers: On the differences between consecutive prime numbe…
Fix irrational numbers $\alpha,\hat\alpha>1$ of finite type and real numbers $\beta,\hat\beta\ge 0$, and let $B$ and $\hat B$ be the Beatty sequences $$ B:=(\lfloor\alpha m+\beta\rfloor)_{m\ge 1}\quad\text{and}\quad\hat…
This is a survey article on the Hardy-Littlewood conjecture about primes in quadratic progressions. We recount the history and quote some results approximating this hitherto unresolved conjecture.
We verify the Hardy-Littlewood conjecture on primes in quadratic progressions on average. The results in the present paper significantly improve those of a previous paper of the authors(arXiv:math.NT/0605563).
We prove near-optimal upper bounds for the odd moments of the distribution of coprime residues in short intervals, confirming a conjecture of Montgomery and Vaughan. As an application we prove near-optimal upper bounds for the average of…
Let $\mathcal{R}_k(n)$ be the number of representations of an integer $n$ as the sum of a prime and a $k$-th power. Define E_k(X) := |\{n \le X, n \in I_k, n\text{not a sum of a prime and a $k$-th power}\}|. Hardy and Littlewood conjectured…
We define a new metric between natural numbers induced by the $\ell_\infty$ norm of their unique prime signatures. In this space, we look at the natural analog of the number line and study the arithmetic function $L_\infty(N)$, which…
Taking $r>0$, let $\pi_{2r}(x)$ denote the number of prime pairs $(p, p+2r)$ with $p\le x$. The prime-pair conjecture of Hardy and Littlewood (1923) asserts that $\pi_{2r}(x)\sim 2C_{2r} {\rm li}_2(x)$ with an explicit constant $C_{2r}>0$.…
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…
It is an open question of Erd\H{o}s as to whether the alternating series $\sum_{n=1}^\infty \frac{(-1)^n n}{p_n}$ is (conditionally) convergent, where $p_n$ denotes the $n^{\mathrm{th}}$ prime. By using a random sifted model of the primes…
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…
The most common difference that occurs among the consecutive primes less than or equal to $x$ is called a jumping champion. Occasionally there are ties. Therefore there can be more than one jumping champion for a given $x$. In 1999 A.…
We prove several results regarding the distribution of numbers that are the product of a prime and a $k$-th power. First, we prove an asymptotic formula for the counting function of such numbers; this generalises a result of E. Cohen. We…
The Bateman--Horn Conjecture predicts how often an irreducible polynomial $f(x) \in \mathbb{Z}[x]$ assumes prime values. We demonstrate that with sufficient averaging in the coefficients of $f$ (viz. exponential in the size of the inputs),…
Quite recently, in [8] the authoor of this paper considered the distribution of primes in the sequence $(S_n)$ whose $n$th term is defined as $S_n=\sum_{k=1}^{2n}p_k$, where $p_k$ is the $k$th prime. Some heuristic arguments and the…
We relate the size of the error term in the Hardy-Littlewood conjectured formula for the number of prime pairs to the $L^{1}$ norm of an exponential sum over the primes formed with the von Mangoldt function.
Consider a system \Psi of non-constant affine-linear forms \psi_1,...,\psi_t: Z^d -> Z, no two of which are linearly dependent. Let N be a large integer, and let K be a convex subset of [-N,N]^d. A famous and difficult open conjecture of…
Legendre's Conjecture is one of the most elegant open problems in Number Theory, which states that there is a prime between consecutive two perfect squares. In this note, we prove the conjecture holds true and also discuss the related…
We study the gaps between consecutive prime numbers directly through Eratosthenes sieve. Using elementary methods, we identify a recursive relation for these gaps and for specific sequences of consecutive gaps, known as constellations.…
One of the questions of distribution of prime numbers is considered in the article. It is shown what error is obtained from the assumption that the asymptotic density of a sequence of primes is a probability. Various forms of an analogue of…
We prove some theorems which give sufficient conditions for the existence of prime numbers among the terms of a sequence which has pairwise relatively prime terms.