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A Chebychev's type of prime number theorem in a short interval II.

Abstract : 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.
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Lou Shituo, Yao Qi. A Chebychev's type of prime number theorem in a short interval II.. Hardy-Ramanujan Journal, Hardy-Ramanujan Society, 1992, Volume 15 - 1992, pp.1 - 33. ⟨10.46298/hrj.1992.124⟩. ⟨hal-01108637⟩

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