Given the advanced results obtained by analytic means in Number Theory, it puzzles me why I don’t recall ever seeing a partial differential equation used to good effect in Number Theory. Is there such?
2026-04-09 11:13:55.1775733235
Is there any PDE that applies specifically to Number Theory?
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One great source of examples is the use of spectral theory and spectral decompositions in automorphic forms. In particular, there are a basis of Maass forms for $\mathbb{H_n}/\text{SL}_n(\mathbb{Z})$. Their existence is very important to the study of automorphic forms, and they are "simply" solutions of some PDE on the relevant surface.
There are books on spectral theory by Iwaniec, papers by Hoffstein (or Sarnak, or many others), and overview books by Goldfeld that document these and some of their properties very nicely.
A different application is contained in Lax's Hyperbolic PDE book. In chapter 9, he shows that the Riemann Hypothesis can be realized as scattering rates of certain automorphic waves, and thus there is a corresponding PDE there as well.