In this paper, we show that $0.969\frac{y}{\log x}\leq\pi(x)-\pi(x-y)\leq1.031\frac{y}{\log x}$, where $y=x^{\theta}, \frac{6}{11}<\theta\leq 1$ with $x$ large enough. In particular, it follows that $p_{n+1}-p_n<\!\!\!0$, where $p_n$ denotes the $n$th prime.
Publié le : 1992-07-04
Classification:
complementary sum.,
number of primes in short intervals,
[MATH]Mathematics [math]
@article{hal-01108637,
author = {Shituo, Lou and Qi, Yao},
title = {A Chebychev's type of prime number theorem in a short interval II.},
journal = {HAL},
volume = {1992},
number = {0},
year = {1992},
language = {en},
url = {http://dml.mathdoc.fr/item/hal-01108637}
}
Shituo, Lou; Qi, Yao. A Chebychev's type of prime number theorem in a short interval II.. HAL, Tome 1992 (1992) no. 0, . http://gdmltest.u-ga.fr/item/hal-01108637/