What are some techniques and theorems of analytic number theory that have proved useful outside of number theory?



Spinoffs of analytic number theory
Look at the list of examples of zeta-functions on Wikipedia, and not all of them are in number theory.


Here are some specific applications of the idea of a zeta-function in other areas of mathematics. 


1. If $G$ is a finitely generated group, let $a\_n$ be the number of subgroups of index $n$ and consider $\zeta\_G(s) = \sum\_{n \geq 1} a\_n/n^s$. An application of the analytic properties of this zeta-function to counting subgroups of $G$ is in Corollary 1.1 of <http://www.icm2006.org/proceedings/Vol_II/contents/ICM_Vol_2_07.pdf>.
2. There is a group-theoretic criterion to find nonisomorphic number fields with the same zeta-function. Sunada was inspired by this idea to discover a way of finding non-isometric Riemannian manifolds with the same spectral zeta-function (same eigenvalues of the Laplacian, with same multiplicities). This led to the first *systematic* way of constructing examples of such pairs of manifolds, and eventually to the first examples of negative answers to the question "Can you hear the shape of a drum?" in the plane. See Section 2.4 of <http://www.mims.meiji.ac.jp/publications/2008/abst00013.pdf>.
3. Suppose random text is produced from a keyboard where the space bar has a fixed probability of being hit and the other keys all have a common probability of being hit. A string of non-space characters separated from other strings by a space at both ends is called a word (the first word may not have a space preceding it). Assume there is more than one non-space key, so there is more than one word of each length. Words with equal length will have equal probability of appearing, and the $j$th most common word -- ties are allowed -- appears with a probability that decays like $j^{\log\_n(1-p)-1}$ up to a bounded scaling factor, where $n > 1$ is the number of non-space keys and $p$ is the common probability of each non-space key being pressed. Because this decay formula is a power function of $j$, the appearances of the words are said to obey a power law. If the probabilities of different non-space keys being hit are not all equal, do the frequencies of the words still obey a power law? Yes, and the proof of that uses estimates on the growth of the partial sums of the coefficients of a *generalized* Dirichlet series. See <http://www.eecs.harvard.edu/~michaelm/postscripts/toit2004a.pdf>, especially the case of irrational log-ratios near the end of the paper.


I'm not sure if it counts, but Hadamard's factorization theorem in complex analysis was directly inspired by the desire to factor $\zeta(s)$ over its zeros.


