论文标题
在一个间隔中存在双素
On the existence of twin prime in an interval
论文作者
论文摘要
令$ s _ {（x，y]} = \ left \ {\ frac {p_n} {p_ {n+1} -2} -2} -2}：〜n \ in I \ right \} $，其中$ i = \ i = \ left \ left \ left \ { \ Mathbb {r} _ {> 0} $。 \ infty}m_α（x，y）> 1-2/y + o（y^{ - 2}）$，用于特殊选择$ y $，我们还为均值降低了：$ \ displayStyle \ lim_ {α\ rightarrow \ infty}m_α（x，x^β）> 1-c/x^β+o（x^{ - β} \ log^{ - 1} x）$，其中常数$ c> 0 $ c> 0 $和$β= 1+c/\ log^2 x $或等效，以$ \ displayStyle \ lim_ {α\ rightarrow \ infty}m_α（x，x^β）$绑定，这满足了在间隔$（x，x，x^β] $中存在双prime的不平等。
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 $M_α(x,y)$ denotes the $α$-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 $\displaystyle \lim_{α\rightarrow \infty}M_α(x,y) > 1 - 2/y + O(y^{-2})$ for a sufficiently large $y$. For a special choice of $y$, we also find a lower bound for the mean: $\displaystyle \lim_{α\rightarrow \infty}M_α(x,x^β)>1-c/x^β+O(x^{-β}\log^{-1} x)$, where the constant $c>0$ and $β= 1+c/\log^2 x$ or equivalently, $x^β=x+cx/\log x+O(x/\log^2 x)$. With $c<2$, the lower bound for $\displaystyle \lim_{α\rightarrow \infty}M_α(x,x^β)$ satisfies the inequality on the existence of a twin prime in the interval $(x,x^β]$.
