Now we are finally ready to see Riemann's result, by plugging in this formula for \(J\) into the Moebius inverted formula for \(\pi\) given by \[\pi(x)=J(x)-\frac{1}{2}J(\sqrt{x})-\frac{1}{3}J(\sqrt[3]{x})-\frac{1}{5}J(\sqrt[5]{x})+\frac{1}{6}J(\sqrt[6]{x})+\cdots\] Riemann did not prove it fully rigorously, and indeed one of the provers of the PNT mentioned taking decades to prove all the statements Riemann made in this one paper, just so he could prove the PNT. Nonetheless, it is certainly Riemann's formula for \(\pi(x)\), and an amazing one:

\[\pi(x)=\sum_{n=1}^\infty \frac{\mu(n)}{n}\left[ Li(x^{1/n})-\sum_{\rho}\left(Li(x^{\rho/n})+Li(x^{\bar{\rho}/n})\right)+\int_{x^{1/n}}^\infty\frac{dt}{t(t^2-1)\ln(t)}\right]\]

It is worth making a few points here.

• If you're wondering where \(\ln(2)\) went, it went to 0 because \(\sum_{n=1}^\infty \frac{\mu(n)}{n}=0\), though this is very hard to prove (in fact, it is a consequence of the PNT).
• Here, \(\rho\) is a zero above the real axis, and \(\bar{\rho}\) is the corresponding one below the real axis.
• In particular, these \(\rho\) are conjectured by the Riemann Hypothesis to all have real part equal to \(1/2\), which would make things particularly tidy.

Now let's see this formula in action.

This graphic shows just how good it can get. Again, notice the waviness, which allows it to approximate \(\pi(x)\) not just once per “step” of the function, but along the steps.

We can also just check out some numerical values.

Just a few of the things which follow from the Riemann Hypothesis (if it is true) or from a natural generalization include:

• The Dirichlet series of the Möbius function really is the multiplicative inverse of the zeta function for lots more complex values than just the real ones we proved it for.
• The value (not just average) of \(\sigma(n)\) has the following bound once \(n\) is big enough: \[\sigma(n)<e^\gamma \ln(\ln(n))\]
• The biggest gap between consecutive prime numbers is not too big (to be precise, it's \(O(\sqrt{p}\ln(p)\)).
• We know exactly what it means for a type of prime to win the “prime races” (assuming a more general version).
• Artin's conjecture on primitive roots follows from a generalization as well.

So can you prove that there are no other zeros other than those on the critical line to contribute to these approximations to \(\pi(x)\)? If so, welcome to the future of number theory!