What are the real world applications of analytic number theory?

Question

I have seen lots of papers on analytic number theory, but I always wonder, what are the uses of it? I don't think showing that the number of primes less than or equal to $x$ is $\frac{x}{\ln(x)}$ will help solve real-world problems. So how did we think of these problems in the first place? It seems how random people would start this up. Can you give me an answer?

Answer 1

Many (most?) mathematicians chose that profession because they found mathematical questions fascinating for their own sake, not necessarily because parts of the subject had real world applications.

Until quite recently, that was true of number theory. Now algebraic number theory is in fact quite useful in cryptography. (Perhaps analytic number theory is somewhat less useful.)

Answer 2

In physics, string theorist are excited about Ramanujan's Theta functions. Here is an excerpt.

The number 24 appearing in Ramanujan's function is also the origin of the miraculous cancellations occurring in string theory. Each of the 24 modes in the Ramanujan function corresponds to a physical vibration of a string. Whenever the string executes its complex motions in space-time by splitting and recombining, a large number of highly sophisticated mathematical identities must be satisfied. These are precisely the mathematical identities discovered by Ramanujan. The string vibrates in ten dimensions because it requires the generalized Ramanujan functions in order to remain self-consistent.