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 and counting points on curves (as in Chapter 15) leads to more complicated series like \(\zeta\), called \(L\)-functions. There is a version of the Riemann Hypothesis for them, too (see the end of the previous subsection). Even without that, they gives truly interesting, strange, and beautiful results. Here is a recent result of interest.
Recall from Example 14.2.3 that the notation \(r_{12}(n)\) should denote the number of ways to write \(n\) as a sum of twelve squares. Here, order and sign both matter, so \((1,2)\) and \((2,1)\) and \((-2,1)\) are all different.
As we let \(p\) run through the set of all prime numbers, the distribution of the fraction \begin{equation*}\frac{r_{12}(p)-8(p^5+1)}{32p^{5/2}}\end{equation*} is precisely as this circular function in the long run:\begin{equation*}\frac{2}{\pi}\sqrt{1-t^2}\end{equation*}
The following graphic is based on one due to William Stein, the original founder and developer of Sage, in personal communication. The higher the number, the closer the values should group to the distribution; change the number of bins in the histogram to see it more clearly.
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!
