Prime number theorem

The prime number theorem states that the prime counting function $\pi(x)$ is asymptotically equivalent to $\frac{x}{\log x}$. At some point it might be interesting to delve further into the proof of this fact, but in this post I just wanted to bring about the connection to zeroes of the Riemann zeta function.

To do this, consider the function $\psi(x)=\sum_{p^r<x} \log p,$ originally defined by Chebyshev. It is not that difficult to show that the prime number theorem is equivalent to $\psi(x)\sim x$. It is more difficult to show the following amazing formula of Von Mangoldt

$$\psi(x)=x-\log(2\pi)-\frac{1}{2}\log(1-x^{-2})-\sum_{\zeta(\rho)=0}\frac{x^\rho}{\rho},$$

where the sum is over non-trivial zeroes of the Riemann zeta function. My mind boggles at the beauty and simplicity of this formula!

Now if every zero $\rho =a +ib$ has real part $a<1$, we can see that $|x^\rho|=|x|^a$, so dividing both sides of the Von Mangoldt formula by $x$, each summand of the series approaches $0$ and with a bit more effort one can show that the whole series approaches $0$. Thus PNT follows simply from showing that there are no zeroes of the form $1+ib$.

One reason I find this formula so beautiful is that by plugging in the mysterious zeroes of $\zeta$ one gets closer and closer approximations to $\psi(x).$ See this page for more on that as well as a neat animation.