相关论文: The Hardy-Littlewood conjectures on the twin prime…

We give a more comrehensive treatment of Chen's double sieve and improve related constants in Goldbach's conjecture and the twin prime problem.

Due to the distribution of primes among integers, we establish an upper bound for the probability $\mathbb{P}_n$ that the Goldbach conjecture fails. Assuming the conjecture holds true for all even number less than $2N$, we prove this…

Let $p_{1}$, ..., $p_{k}$ be the first $k$ odd primes in succession. Let $n$ be an even integer such that $n > p_{k}$. We conjecture that if none of $n - p_{1}$, ..., $n - p_{k}$ are prime, then at least one of them has a prime factor which…

We provide two new proofs of the infinitude of prime numbers, using the additive Ramsey-theoretic result known as Folkman's theorem (alternatively, one can think of these proofs as using Hindman's theorem). This adds to the existing…

We give a large sieve type inequality for functions supported on primes. As application we prove a conjecture by Elliott, and give bounds for short character sums over primes. The proves uses a combination of the large sieve and the Selberg…

We study the number of primes with a given primitive root and in an arithmetic progression under the assumption of a suitable form of the generalized Riemann Hypothesis. Previous work of Lenstra, Moree and Stevenhagen has given asymptotics…

This paper is a part of our programme to generalise the Hardy-Littlewood method to handle systems of linear questions in primes. This programme is laid out in our paper Linear Equations in Primes [LEP], which accompanies this submission. In…

In this paper, we develop the method of circle of partitions and associated statistics. As an application we prove conditionally the binary Goldbach conjecture. We develop a series of steps to prove the binary Goldbach conjecture in full.…

We prove that every odd number $N$ greater than 1 can be expressed as the sum of at most five primes, improving the result of Ramar\'e that every even natural number can be expressed as the sum of at most six primes. We follow the circle…

Myriad articles are devoted to Mertens's theorem. In yet another, we merely wish to draw attention to a proof by Hardy, which uses a Tauberian theorem of Landau that "leads to the conclusion in a direct and elegant manner". Hardy's proof is…

For earlier considered our sequence A166944 in [4] we prove three statements of its connection with twin primes. We also give a sufficient condition for the infinity of twin primes and pose several new conjectures; among them we propose a…

This is a kind of survey on properties of correlations of two very general arithmetic functions, mainly from the point of view of Ramanujan expansions. In fact, our previous papers on these links had, as a focus, the "Ramanujan…

The well-known Hardy--Ramanujan inequality states that if $\omega(n)$ denotes the number of distinct prime factors of a positive integer $n$, then there is an absolute constant $C>0$ such that uniformly for $x\ge2$ and $k\in\mathbb{N}$,…

We introduce a full binary directed tree structure to represent the set of natural numbers, further categorizing them into three distinct subsets: pure odd numbers, pure even numbers, and mixed numbers. We adopt a binary string…

For a fixed polynomial $\Delta$, we study the number of polynomials $f$ of degree $n$ over $\mathbb F_q$ such that $f$ and $f+\Delta$ are both irreducible, an $\mathbb F_q[T]$-analogue of the twin primes problem. In the large-$q$ limit, we…

Dickson conjectured that a set of polynomials will take on infinitely many simultaneous prime values. Later others, such as Hardy and Littlewood, gave estimates for the number of these primes. In this article we look at this conjecture,…

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…

Let $\Lambda(n)$ be the von Mangoldt function, and let $[t]$ be the integral part of real number $t$. In this note, we prove that for any $\varepsilon>0$ the asymptotic formula $$ \sum_{n\le x} \Lambda\Big(\Big[\frac{x}{n}\Big]\Big) =…

We consider the ternary Goldbach problem with two prime variables of the form $k^2+m^2+1$ and find an asymptotic formula for the number of its solutions.

This paper introduces a new method to find the next prime number after a given prime ${P}$. The proposed method is used to derive a system of inequalities, that serve as constraints which should be satisfied by all primes whose successor is…