What are some techniques and theorems of analytic number theory that have proved useful outside of number theory?
[Math] Spinoffs of analytic number theory
analytic-number-theorynt.number-theory
Related Solutions
Yes, there is a theorem to this effect by Takeuti given in his book "Two applications of logic to mathematics". He shows roughly that complex analysis can be developed in a conservative extension of Peano arithmetic.
(C) Recently applied model theorists have touched many areas of algebra, algebraic geometry, number theory and even analysis structures.
(1) Exponential fields:
Schanuel's conjecture is a conjecture made by Stephen Schanuel in the 1960s:
Given any $n$ complex numbers $z_1,\dots,z_n$ which are linearly independent over the rational numbers $\mathbb{Q}$, the extension field $\mathbb{Q}(z_1,\dots,z_n, \exp(z_1),\dots,\exp(z_n))$ has transcendence degree of at least $n$ over $\mathbb{Q}$.
In 2004, Boris Zilber systematically constructs exponential fields $K_{\exp}$ that are algebraically closed and of characteristic zero, and such that one of these fields exists for each uncountable cardinal. Zilber axiomatises these fields and by using the Hrushovski's construction and techniques inspired by work of Shelah on categoricity in infinitary logics, proves that this theory of "pseudo-exponentiation" has a unique model in each uncountable cardinal. See here and here for more.
(2) Polynomial dynamics:
The connection between algebraic dynamics and the model theory of difference fields was first noticed by Chatzidakis and Hrushovski. A series of three papers entitled "Difference fields and descent in algebraic dynamics". It seems that the first-order theories of algebraically closed difference fields where the automorphism is "generic" are quite nice. See here for more result by Scanlon and Alice Medvedev.
(3) Diophantine geometry:
Hrushovski, Scanlon and their students have worked on model theory and its application in Diophantine geometry. See here for information about applications of model theory in Diophantine geometry.
(4) Algebraic geometry:
The Mordell-Lang conjecture for function fields: Let $k_0\subset K$ be two distinct algebraically closed fields. Let $A$ be an abelian variety defined over $K$, let $X$ be an infinite subvariety of $A$ defined over $K$ and let $\Gamma$ be a subgroup of "finite rank" of $A(K)$. Suppose that $X\cap \Gamma$ is Zariski dense in $X$ and that the stabilizer of $X$ in $A$ is finite. Then there is a subabelian variety $B$ of $A$ and there are $S$, an abelian variety defined over $k_0$, $X_0$ a subvariety of $S$ defined over $k_0$, and a bijective morphism $h$ from $B$ onto $S$, such that $X=a_0 + h^{-1}(X_0)$ for some $a_0$ in $A$.
This theorem is proved by Hrushovski in 1996, see here. For more see this book.
(5) Number theory:
For example see the recent works of Jonathan Pila.
(6) Analysis:
Traditionally model theory is consistent with algebra. But recently, model theorists have been interested in continuous structures that appears in analysis, for example Banach spaces. For more see here.
Model theory has many other application in other fields of mathematics, such as geometric group theory, differential algebra, Berkovich spaces (see recent works of Hrushovski, Loeser, Poonen here and here), approximate groups, etc. (for more see here, here, here and here )
Note: Model theorists have many important and interesting problems in their fields and I believe that the goal of model theory is not necessary to solve the problems of the other fields!
Best Answer
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.
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.
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.
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.