论文标题
在质量差距上有效的指数界限
Effective exponential bounds on the prime gaps
论文作者
论文摘要
在过去的50年中，已经获得了第一个Chebyshev功能上的大量有效指数界限$ \ vartheta（x）$。具体来说，我们将对形式的有效指数界\ [| \ vartheta（x）-x |感兴趣。 <a \; x \;（\ ln x）^b \; \ exp \ left（-c \; \ sqrt {\ ln x} \ right）; \ qquad（x \ geq x_0）。 \]此处，我们应将$ \ vartheta（x）$上的这些有效界限转换为prime差距上的有效指数界限$ g_n = p_n = p_ {n+1} -p_n $。具体来说，我们将建立形式的许多有效的指数界限\ [{g_n \ vy p_n} <{2a \;（\ ln p_n）^b \; \ exp \ left（-c \; \ sqrt {\ ln p_n} \ right）\ vos 1- a \;（\ ln p_n）^b \; \ exp \ left（-c \; \ sqrt {\ ln p_n} \ right）}; \ qquad（x \ geq x_*）; \]和\ [{g_n \ vos p_n} <3a \;（\ ln p_n）^b \; \ exp \ left（-c \; \ sqrt {\ ln p_n} \ right）; \ qquad（x \ geq x_*）; \]对于一些有效的计算$ x _*$。指数因子的明确存在，具有已知系数和已知的有效性范围，这使这些边界特别有趣。
Over the last 50 years a large number of effective exponential bounds on the first Chebyshev function $\vartheta(x)$ have been obtained. Specifically we shall be interested in effective exponential bounds of the form \[ |\vartheta(x)-x| < a \;x \;(\ln x)^b \; \exp\left(-c\; \sqrt{\ln x}\right); \qquad (x \geq x_0). \] Herein we shall convert these effective bounds on $\vartheta(x)$ into effective exponential bounds on the prime gaps $g_n = p_{n+1}-p_n$. Specifically we shall establish a number of effective exponential bounds of the form \[ {g_n\over p_n} < { 2a \;(\ln p_n)^b \; \exp\left(-c\; \sqrt{\ln p_n}\right) \over 1- a \;(\ln p_n)^b \; \exp\left(-c\; \sqrt{\ln p_n}\right)}; \qquad (x \geq x_*); \] and \[ {g_n\over p_n} < 3a \;(\ln p_n)^b \; \exp\left(-c\; \sqrt{\ln p_n}\right); \qquad (x \geq x_*); \] for some effective computable $x_*$. It is the explicit presence of the exponential factor, with known coefficients and known range of validity for the bound, that makes these bounds particularly interesting.
