Pi hiding in prime regularities

There are only so many times that one can use the phrase "This is my favorite piece of math" before it starts to lose meaning.  So instead I'll restrain myself and say this is my favorite piece of number theory.

A surprising number of formulas relate things the Riemann zeta function, as well as some of its cousins, to pi.  Most notably the values of zeta on positive even integers.  Here I explain one example from this family, where it's most clear how circles are involved.  What I think is fascinating is that you could set out to find a formula for pi by counting lattice points in the way described here, with no intention of talking about prime number or multiplicative functions or any of that, and moreover the final formula itself seems to be clean of primes, but nevertheless the natural course of exploration seems to have this gravitational pull towards patterns in primes.

The fact that similar formulas arise from the zeta function itself, which is known to encode the information regarding the distribution of primes, is one more piece of evidence that this connection is not just a coincidence.