On the existence of twin prime in an interval

TL;DR

This paper investigates the conditions under which twin primes exist within a given interval by analyzing the behavior of a specific mean derived from prime ratios, providing new bounds related to twin prime occurrence.

Contribution

It introduces a novel criterion involving the limit of an alpha-power mean of prime ratios that implies the existence of twin primes in an interval.

Findings

Establishes a condition linking the mean limit to twin prime existence.

Provides a lower bound for the mean in specific intervals.

Suggests that the mean's behavior supports twin prime conjecture in certain ranges.

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 $\displaystyle \lim_{\alpha \rightarrow \infty}M_{\alpha}(x,y) > 1 - 2/y + O(y^{-2})$ for a sufficiently large $y$. For a special choice of $y$, we also find a lower bound for the mean: $\displaystyle \lim_{\alpha \rightarrow \infty}M_{\alpha}(x,x^\beta)>1-c/x^\beta+O(x^{-\beta}\log^{-1} x)$, 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 $\displaystyle \lim_{\alpha \rightarrow \infty}M_{\alpha}(x,x^\beta)$ satisfies the inequality on the existence of a twin prime in the interval $(x,x^\beta]$.