A Very Elementary Proof of Prime Number Theorem

In the context below, \(p\) will denote prime numebers always.

As shown by Zhihu user Pandora Eartha that \[ \sum_{p\le n} \frac{1}{p}\sim \ln \ln n. \] Assume the prime number distribution function \(f(x)\), then \[ \int^x \frac{f(x’)}{x’}dx’\sim \sum_{p\le x} \frac{1}{p} \sim \ln \ln x. \] Differentiating on both sides yields \[ \frac{f(x)}{x} \sim \frac{1}{x\ln x} \] and in turn suggests that \(f(x)\sim 1/\ln x\). Therefore \[ \pi(x) \sim \int^n f(x)dx \sim \int^n \frac{1}{\ln x}dx = \mathrm{Li}(n) \] which is what we want.

The lemma we used can be found at https://www.zhihu.com/question/1918807288170386258/answer/1945782129632974139. Nevertheless, I will post a quick version of their proof. \[ \begin{aligned} \ln\ln n \sim \ln \sum_{x\le n}\frac{1}{x} \sim \ln\prod_{p\le n}\sum_{i=0}^\infty \frac{1}{p^i}
=\ln \prod_{p\le n} \frac{1}{1-1/p}
\sim \sum_{p\le n}\ln (1+1/p)
\sim \sum_{p\le n} \frac{1}{p}. \end{aligned} \] This could give a more precise estimate, but is beyond the scope of this short post.