I am really interested in the arts of topology and analytic number theory, so I naturally acquired interest in the potential connections between them. Unlike the algebraic number theory and ongoing arithmetic topology, it seems that topology did not find much application to problems in analytic number theory, say distribution of primes, additive properties, etc. Only interesting paper I found is Furstenburg’s proof of the infinitude of primes using topological arguments, but that was from early 1950.
So my question is that do you know any interesting connection between topology and analytic number theory? Do the set of all primes or natural numbers exhibit some interesting topology (definitely not discrete topology)?