On the existence of twin prime in an interval

Abstract

Let $S_{(x,y]}=\left\{\frac{p_n}{p_{n+1}-2}:n\in I\right\}$, where $I=\left\{n:x<p_n\le y\right\}$, $p_n$ is the $n$-th prime and $x,y\in {\mathbb{R}}_{>0}$. If $M_{\alpha }(x,y)$ denotes the $\alpha$-power mean of the elements of $S_{(x,y]}$, it is shown that the existence of a twin prime pair in $(x,y]$ is implied if $\underset{\alpha \to \infty }{\lim }{M}_{\alpha }(x,y)>1-2/y+O(y^{-2})$ for a sufficiently large $y$. For a special choice of $y$, we also find a lower bound for the mean: $\underset{\alpha \to \infty }{\lim }{M}_{\alpha }(x,x^{\beta })>1-c/x^{\beta }+O(x^{-\beta }{\log }^{-1}x)$, where the constant $c>0$ and $\beta =1+c/{\log }^{2}x$ or equivalently, $x^{\beta }=x+cx/\log x+O(x/{\log }^{2}x)$. With $c<2$, the lower bound for $\underset{\alpha \to \infty }{\lim }{M}_{\alpha }(x,x^{\beta })$ satisfies the inequality on the existence of a twin prime in the interval $(x,x^{\beta }]$.

arXiv comment6 pages; abstract and introduction are slightly modified; for better proof of Lemma 1, Theorem 4 is replaced and corresponding Corollary 1 is added; proofs of theorems and lemmas are provided with more details; results remain the same