Thank you for this. I've favorited this comment so that I can read on each of these to sate my curiosity. Now I'm off to search for accessible resources for these topics for those of us non-mathematicians ;-)
I think once you understand how to apply analytic continuation to the problem its relation to primes is much more apparent; even without a full understanding of the history.
https://en.m.wikipedia.org/wiki/Analytic_continuation