Hardy Ramanujan Journal |

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 $p_{n+1}-p_n<\!\!\!<p_n^{6/11+\varepsilon}$ for any $\varepsilon>0$, where $p_n$ denotes the $n$th prime.

Source : oai:HAL:hal-01108637v1

Volume: Volume 15

Published on: January 1, 1992

Submitted on: March 3, 2015

Keywords: complementary sum.,number of primes in short intervals,[MATH] Mathematics [math]

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