The set $\{\frac{p}{q}\}$ (with $p,q$ primes) is dense in $[0,1]$?

Question

Question: Put $\Lambda=\{\frac{p}{q}: p<q, \text{both p, q are prime}\}$. Is $\Lambda$ dense in $[0,1]$?

This problem appeared in my attempt to simplify the proof of a Fourier multiplier theorem.

As a corollary of weak twin prime conjecture, I know that $1\in \overline{\Lambda}$.

Can any one offer me some references?

Answer 1

Notation:

• $\mathbb{P}$ the set of primes numbers.
• $A=\left\{\frac{p}{q}: p,q\in \mathbb{P}\right\}$.
• $\pi(x):=\text{Card}\left\{ p\in \mathbb{P}: p\leqslant x\right\}$

Theorem:

• Prime number theorem: $$\displaystyle \pi(x)\underset{+\infty}{\sim}\frac{x}{\log x}$$ In words the prime number theorem provides a way to approximate the number of primes less than or equal to a given number $x$.

Problem

• We need to prove that $A$ is dense on $[0;1]$. Moreover is dense on $[0;+\infty[ \supseteq [0;1]$.

HINT:

• Given positive integers $a$ and $b$, then examine the limit of the sequence $(a_{n})_{n=1}^{+\infty}$ with $a_{n}:=\frac{p_{an}}{p_{bn}}$.