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

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up vote 11 down vote accepted

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.

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Sieve Theory seems like an example.

Originally developped for question related to twin-primes and Golbach-like questions, and other clearly number theoretic questions, such techniques are meanwhile used to address
other types of problems as well.

To illustrate this, as one example, let me mention the following result (Corollary 9.7) in a preprint of Kowalski (The principle of the large sieve):

Let $G$ be the mapping class group of a closed orientable surface of genus $g > 1$, let $S$ be a finite symmetric generating set of $G$ and let $(X_k)$, $k > 1$, be the simple left-invariant random walk on $G$. Then the set $X \subset G$ of non-pseudo-Anosov elements is transient for this random walk.

To get a more detailed impression, this preprint or (even better) the book by the same author, The Large Sieve and its Applications (CUP, 2008), into which this preprint developped, would be a good source.

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The Riemannian Zeta function is sometimes important in physics, it occurs both in thermodynamics and in quantum field theory. See for example: http://en.wikipedia.org/wiki/Riemann_zeta_function#Applications

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We can even find applications of analytic number theory techniques outside of mathematics. Think e.g. on the relations between Riemann zeta function and Casimir effect (physics) or Zipf's law (statistics).

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Arithmetic geometry is still arguable number theory I suppose, but this is pretty geometric: in recent years, there have been applications of the Circle Method to the study of rational curves on varieties over finite fields (e.g. http://www.math.columbia.edu/~thaddeus/theses/2011/pugin.pdf).

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