An example described in the article, and which comes from the same branch of analytic number theory, is Daniel Shiu's result that you can find arbitrarily long

An example described in the article, and which comes from the same branch of analytic number theory, is Daniel Shiu's result that you can find arbitrarily long strings of primes in "any" arithmetic progression. (By "any" I mean any arithmetic progression a (mod m) where a and m don't share any common prime factor.) So, for example, there is a string of a billion consecutive primes, each of whose decimal expansions ends in a billion ones. In principle you could use the method to find it, possibly the first example would be around 10^(10^10) or so. The gateway to all of this is the quantitative form of Dirichlet's theorem on prime numbers in arithmetic progressions: given coprime a and m, there are infinitely many primes which are a (mod m). (So, in other words, Shiu's result with "just one prime".) In this case, we know asymptotically how many such primes there are, and we can get upper bounds on the error. The proof is highly non-constructive, and uses complex analysis applied to the so-called "L-functions". Using a variety of trickery ("sieve methods", "the circle method", etc.) this can then be leveraged into other interesting results.