Conditional Bounds for Prime Gaps with Applications

TL;DR

This paper explores bounds and properties of prime gaps, prime counting functions, and related conjectures, proposing new inequalities and conjectures that connect prime distribution with square roots and fractional parts.

Contribution

It introduces new bounds on prime gaps, proves inequalities involving prime counting functions, and conjectures about accumulation points of fractional parts of square roots of primes.

Findings

Proves that $d_n^2 < 2p_{n+1}$ for all $n extgreater{}0$

Shows $rac{d_n}{\sqrt{p_n}} o 0$ as $n o\infty$

Conjectures about accumulation points of fractional parts of $\sqrt{p_n}$

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 $x$.