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.
This is a collection of some of my lesser known, but nonetheless appealing, problems. Its main focus is on problems concerning the representation of integers as sums or products of integers from a given sequence $\{a_n\}$.