Show that $p_n^{1-\epsilon}\le n$ using PNT

Solution 1:

$\color{red}{\text{1. One way}}$

I will use 2 results

• PNT $$\lim\limits_{n\rightarrow\infty}\frac{\pi(n)\ln{(n)}}{n}=1 \Rightarrow \lim\limits_{n\rightarrow\infty}\frac{n}{\pi(n)\ln{(n)}}=1 \tag{1}$$
• and $$\lim\limits_{n\rightarrow\infty}\frac{p_n}{n\ln{(n)}}=1 \tag{2}$$

Proposition 1.1 $$\lim\limits_{n\rightarrow\infty} \frac{\ln{(n)}}{\ln{(p_n)}}=1$$

$\{p_n\}$ is a subsequence of $\{n\}$, thus, from $(1)$, $$\lim\limits_{p_n\rightarrow\infty}\frac{\pi(p_n)\ln{(p_n)}}{p_n}=1 \Rightarrow \lim\limits_{n\rightarrow\infty}\frac{\pi(p_n)\ln{(p_n)}}{p_n}=1 \Rightarrow ...$$ because $\pi(p_n)=n$ $$...\lim\limits_{n\rightarrow\infty}\frac{n\ln{(p_n)}}{p_n}=1 \tag{3}$$ Now $$\lim\limits_{n\rightarrow\infty} \frac{\ln{(n)}}{\ln{(p_n)}}= \lim\limits_{n\rightarrow\infty} \left(\frac{n\ln{(n)}}{p_n}\cdot\frac{p_n}{n\ln{(p_n)}}\right)=\\ \lim\limits_{n\rightarrow\infty} \left(\frac{n\ln{(n)}}{p_n}\right)\cdot \lim\limits_{n\rightarrow\infty} \left(\frac{p_n}{n\ln{(p_n)}}\right)\overset{(2)(3)}{=}1$$

Proposition 1.2 For large enough $n$ $$p_n^{1-\varepsilon}<n$$

From $$\lim\limits_{n\rightarrow\infty} \frac{\ln{(n)}}{\ln{(p_n)}}=1$$ using the definition of limit, $\forall\varepsilon >0, \exists N(\varepsilon)\in\mathbb{N}$ s.t. $\forall n> N(\varepsilon)$ $$\left|\frac{\ln{(n)}}{\ln{(p_n)}}-1\right|<\varepsilon \Rightarrow 1-\varepsilon <\frac{\ln{(n)}}{\ln{(p_n)}}< 1+\varepsilon \Rightarrow \\ (1-\varepsilon)\ln{(p_n)} <\ln{(n)}< (1+\varepsilon)\ln{(p_n)} \Rightarrow \\ \ln{(p_n)^{(1-\varepsilon)}} <\ln{(n)}< \ln{(p_n)^{(1+\varepsilon)}} \Rightarrow ...$$ $e^x$ is ascendng, thus $$... p_n^{1-\varepsilon} <n< p_n^{1+\varepsilon} $$

$\color{red}{\text{2. Another way}}$

Using Vallée-Poussin, for large enough $x$ $$\pi(x)>\frac{x}{\ln(x)-(1-\varepsilon)}>\frac{x}{\ln(x)}$$ Let's show that for large enough $x$ we also have $$\frac{x}{\ln(x)}>x^{1-\varepsilon}$$ which is the same as showing $$\frac{x^{\varepsilon}}{\ln{x}}>1$$ for large $x>0$.

Propositions 2.1 Function $f(x)=\frac{x^{\varepsilon}}{\ln{x}}$ is ascending for large $x>0$.

Because $$f'(x)=\frac{x^{\varepsilon-1} (\varepsilon \ln{x}-1)}{\ln^2{x}}>0 \iff \varepsilon \ln{x}-1>0 \Rightarrow x> e^{\frac{1}{\varepsilon}}$$

Propositions 2.2 $\lim\limits_{x\rightarrow \infty}f(x) \rightarrow \infty$

If we assume it is bounded by a large $\alpha>0, \forall x>1$ and we know $\ln{x}$ is ascending $$\frac{x^{\varepsilon}}{\ln{x}} < \alpha \iff 1<x^{\varepsilon}< \alpha \ln{x} \iff \color{red}{0}<\varepsilon<\frac{\ln{\alpha}}{\ln{x}}+\frac{\ln{\ln{x}}}{\ln{x}}\rightarrow \color{red}{0}, x\rightarrow\infty$$ which is a contradiction.

So, for large $x$ we have $$\pi(x)>x^{1-\varepsilon}$$ which means, for large $n$ we have $$n=\pi(p_n)>p_n^{1-\varepsilon}$$