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Are there any good links between number theory and differential equations?

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Arguably complex analysis is all about solutions of a system of PDEs (Cauchy-Riemann equations), and complex analysis is *very* useful in number theory.

Or look up p-adic differential equations.
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Take a look in N. Katz paper “Exponential Sums over Finite Fields and Differential Equations over the Complex Numbers: Some Interactions”. (This is a summary of his book Exponential Sums and Differential Equations).

This is really one of the most incredible pieces of mathematics that I know. Exponential sums are one of the main objects in modern analytic number theory. And being able to understand them using differential equations (and passing through a lot of algebraic geometry!) is absolutely breathtaking.
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Modular forms can show up as eigenfunctions of Laplace operators, thus one can do number theory via spectral analysis of certain differential operators on suitable complex manifolds.
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Yes bifurcation problems for PDEs like the wave or heat equations frequently result in Diophantine equations due to the presence of the Laplacian if one works on e.g. the torus. The solutions of these Diophantine equations can correspond to bifurcating solutions of the PDE.
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Look up the Dickman function. It satisfies a differential delay equation and gives an asymptotic result for the number of integer free of large prime factors.
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The heat equation is at the intersection between those two subjects
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There's a lot of work on p-adic differential equations. I think Kedlaya just put out a book on that. This usually goes by names such as D-modules, crystals, connections/stratifications, and so on which aren't inmediately recognizable as DEs.

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