Related papers: Analog of the Skewes number for twin primes

The results of the computer investigation of the sign changes of the difference between the number of twin primes $\pi_2(x)$ and the Hardy--Littlewood conjecture $C_2\Li_2(x)$ are reported. It turns out that $d_2(x)=\pi_2(x) - C_2\Li_2(x)$…

The results of the computer hunt for the primes of the form $q = m^2+1$ up to $10^{20}$ are reported. The number of sign changes of the difference $\pi_q(x) - \frac{C_q}{2}\int_2^x{du \over \sqrt{u}\log(u)}$ and the error term for this…

For any positive integer r, let pi_{2r}(x) denote the number of prime pairs (p, p+2r) with p not exceeding (large) x. According to the prime-pair conjecture of Hardy and Littlewood, pi_{2r}(x) should be asymptotic to 2C_{2r}li_2(x) with an…

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 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…

Let $m$ and $n$ be positive integers with $m,n \geq 2$. The second Hardy-Littlewood conjecture states that the number of primes in the interval $(m,m+n]$ is always less than or equal to the number of primes in the interval $[2,n]$. Based on…

The twin prime conjecture asserts that there are infinitely many pairs of primes that differ by two. While recent advances have improved our understanding of bounded prime gaps, the conjecture remains unresolved. This paper refines the…

In the article we establish the Hardy-Littlewood inequality $ \pi (x + y) \leq \pi (x) + \pi (y) $. We also prove that the naturally ordered primes $p_1=2,p_2=3,p_3=5,p_4=7,\dots$ satisfy the inequality $ p_ {a + b}> p_a + p_b $ for all $a,…

We adopt a physically motivated empirical approach to the characterisation of the distributions of twin and triplet primes within the set of primes, rather than in the set of all natural numbers. Remarkably, the occurrences of twins or…

In 1922 Hardy and Littlewood proposed a conjecture on the asymptotic density of admissible prime k-tuples. In 2011 Wolf computed the "Skewes number" for twin primes, i.e., the first prime at which a reversal of the Hardy-Littlewood…

The second Hardy-Littlewood conjecture asserts that the prime counting function $\pi(x)$ satisfies the subadditive inequality \begin{align*} \pi(x+y)\leqslant \pi(x)+\pi (y) \end{align*} for all integers $x,y\geqslant 2$. By linking the…

We study the prime pair counting functions $\pi_{2k}(x),$ and their averages over $2k.$ We show that good results can be achieved with relatively little effort by considering averages. We prove an asymptotic relation for longer averages of…

The prime counting function inequality $\pi(x+y) < \pi(x)+\pi(y)$, which is known as Hardy-Littlewood conjecture, has been established for a variety of cases such as $ \delta x \leq y \leq x$, where $0< \delta \leq 1$, and $x \leq y\leq x…

We establish, utilizing the Hardy-Littlewood Circle Method, an asymptotic formula for the number of pairs of primes whose differences lie in the image of a fixed polynomial. We also include a generalization of this result where differences…

Let $V(T)$ denote the number of sign changes in $\psi(x) - x$ for $x\in[1, T]$. We show that $\liminf_{\;T\rightarrow\infty} V(T)/\log T \geq \gamma_{1}/\pi + 1.867\cdot 10^{-30}$, where $\gamma_{1} = 14.13\ldots$ denotes the ordinate of…

The paper substantiates the conjecture of the asymptotic behavior of the largest distance between consecutive primes: $sup_{p_i \leq x}(p_{i+1}-p_i) \sim 2e^{-\gamma} \log^2(x)$, where $\gamma$ is the Euler constant. The Hardy-Littlewood…

We present an algorithm analogous to the sieve of Eratosthenes that produces the list of twin primes. Next, we count the number of twin primes resulting from the construction with two different heuristic arguments. The first method is…

The second Hardy-Littlewood conjecture, that $\pi(x)+\pi(y) \geq \pi(x+y)$ for integers $x$ and $y$ with $\min\{x,y\}\geq 2$, was formulated in 1923. It continues to attract attention to this day, almost 100 years later. In 1975 Udrescu…

We give an estimation of the existence density for the $2d$ different primes by using a new and simple algorithm for getting the $2d$ different primes. The algorithm is a kind of the sieve method, but the remainders are the central numbers…

We establish, via a formal/heuristic Fourier inversion calculation, that the Hardy-Littlewood twin prime conjecture is equivalent to an asymptotic formula for the two-point correlation function of Riemann zeros at a height $E$ on the…