标题： 包含素数的有效短区间 显示英文标题

标题： Effective short intervals containing primes

摘要： 95年前，Hoheisel证明了在子线性区间\[ \left[x, x+x^{1-{1\over 33000}}\right] \qquad \hbox{for $x$ sufficiently large}. \]中存在素数。这被Heilbronn改进，证明了在区间\[ \left[x, x+x^{1-{1\over 250}}\right] \qquad \hbox{for $x$ sufficiently large}. \]中存在素数。最近，Baker、Harman、Pintz证明了在区间\[ \left[x, x+ x^{1-{19\over 40}}\right] \qquad \hbox{for $x$ sufficiently large}. \]中存在素数。在本文中，我将尽可能使这些陈述变得有效。 具体来说，我将证明\[ \forall n \geq 4, \qquad\forall x \geq \exp(\exp(33)), \qquad \hbox{there are primes in the interval} \left[x, x+ x^{1-{1\over n}}\right]; \] \[ \forall n \geq 91, \qquad\forall x \geq [90^{90}]^{n/(n-90)} , \qquad \hbox{there are primes in the interval} \left[x, x+ x^{1-{1\over n}}\right]. \] 此外，\[ \forall n \geq 106, \qquad\forall x \geq 1, \qquad \hbox{there are primes in the interval} \left[x, x+ x^{1-{1\over n}}\right]. \]特别是这一最后的观察使得Hoheisel和Heilbronn的结果完全明确且有效。 这个（相对）具体的观察可以以各种方式扩展和推广。 显示英文摘要

摘要： 95 years ago Hoheisel proved the existence of primes in the sub-linear interval \[ \left[x, x+x^{1-{1\over 33000}}\right] \qquad \hbox{for $x$ sufficiently large}. \] This was improved by Heilbronn, proving existence of primes in the interval \[ \left[x, x+x^{1-{1\over 250}}\right] \qquad \hbox{for $x$ sufficiently large}. \] More recently Baker, Harman, Pintz proved existence of primes in the interval \[ \left[x, x+ x^{1-{19\over 40}}\right] \qquad \hbox{for $x$ sufficiently large}. \] In the present article I will, to the extent possible, make some of these statements effective. Specifically, among other things, I shall show that \[ \forall n \geq 4, \qquad\forall x \geq \exp(\exp(33)), \qquad \hbox{there are primes in the interval} \left[x, x+ x^{1-{1\over n}}\right]; \] \[ \forall n \geq 91, \qquad\forall x \geq [90^{90}]^{n/(n-90)} , \qquad \hbox{there are primes in the interval} \left[x, x+ x^{1-{1\over n}}\right]. \] Furthermore \[ \forall n \geq 106, \qquad\forall x \geq 1, \qquad \hbox{there are primes in the interval} \left[x, x+ x^{1-{1\over n}}\right]. \] In particular this last observation makes both the Hoheisel and Heilbronn results fully explicit and effective. This (relatively) specific observation can be extended and generalized in various manners.