Conditional Bounds for Prime Gaps with Applications

Abstract

We posit that $d_n^2 < 2p_{n+1}$ holds for all $n\geq 1$, where $p_n$ represents the $n$th prime and $d_n$ stands for the $n$th prime gap i.e. $d_n := p_{n+1} - p_n$. Then, the presence of a prime between successive perfect squares, as well as the validity of $\Delta_n := \sqrt{p_{n+1}} - \sqrt{p_n} < 1$ are derived. Next, $\pi(x)$ being the number of primes $p$ up to $x$, we deduce $\pi(n^2-n) < \pi(n^2) < \pi(n^2+n)$ $(n\geq 2)$. In addition, a proof of $\pi((n+1)^k) - \pi(n^k) \geq \pi(2^k)$ \ $(k\geq 2, n\geq 1)$ is worked out. The vanishing nature of $\Delta_n$ as $n$ goes to infinity is set, and used afterwards to achieve both $\displaystyle{\lim_{n\rightarrow\infty}d_n/\sqrt{p_n} = 0}$ and the twin prime conjecture. Also, question about the estimate $p_n < 2j_n^2 \ (n\geq 6)$, where $j_n$ counts the twin prime pairs up to $p_n$, is raised. Finally, we put forward the conjecture that any rational number $r$ $(0\leq r \leq 1)$ represents an accumulation point of the sequence $\left(\{\sqrt{p_n}\}\right)_{n\geq 1}$, where $\{x\}$ acts for the fractional part of $x$.