Let's see just a little more of the future of number theory. The Riemann zeta function and counting primes is truly only the beginning of research in modern number theory.

For instance, research in finding points on curves leads to more complicated series like \(\zeta\), called \(L\)-functions. There is a version of the Riemann Hypothesis for them, too!

And they gives truly interesting, strange, and beautiful results. Here is a result from the past few years.

Let \(r_{12}(n)\) denote the number of ways to write \(n\) as a sum of twelve squares, like we did \(r(n)\) the number of ways to write as a sum of two squares. Here, order and sign both matter, so \((1,2)\) and \((2,1)\) and \((-2,1)\) are all different.

What an amazing result. These ideas are at the forefront of all types of number theory research today, and my hope is that you will enjoy exploring more of it, both with paper and pencil and using tools like Sage!