Great! Since this has somehow bubbled to the top, I have yet-another occasion for a not-entirely-selfish rant! :)
Although @WillieWong's answer was a few years ago, it does point (if a bit obscurely) in a significant direction, for example. Immediately, one should note that there is no known (to me, and ... I care) proof of RH that immediately/simply uses PDE ideas on modular curves.
But those ideas, from Haas, Hejhal, Faddeev, Colin-de-Verdiere, and Lax-Phillips, do solidly establish a connection of spectral properties of (certain "customized" extensions of) Laplacians on modular curves and related canonical objects.
A more obviously legit example of interaction of number-theoretic "physical objects" and "PDE" was Milnor's example of two tori with the same spectrum for the Laplacian. But, yes, that wasn't really about PDE.
A problem with reporting modern relevance of PDE to number theory is that there are "complicating" additions, ... :) ... often under-the-radar amounting to things stronger than Lindelof Hypothesis, if not actually the Riemann Hypothesis.
Rather than recap things better documented elsewhere (many peoples' arXiv preprints, my own vignettes and talks various places, ...) please forgive my returning to the homily that "PDE" are merely assertions of relations, as Newton intuited for the planets, and others have observed/inferred for many more things.
(Any hysterically provincial remarks about turf, or "specialties-as-ignorance-of-other" are obviously toxic... despite their dangerous prevalence and popularity...)
Thus, srsly, people, "PDE" means "a kind of condition on functions..." ... If people weren't so caught up in ... oop, sorry, the kind of people do get caught up in... :) ... it'd be obvious that "infinitesimal" conditions would be natural...
Thus, an explanatory but not really useful answer is, that we seem to see that This is not related to That because the respective proprietores have no vested interest in letting on that anyone else could ... perform their guild's function.
(Yes, it is informative to review Europe's late-renaissance guild-culture...)
And, as in many rants, I wish to reassure everyone by my disclaimer, "wait, what was the question, ... again...? " :)
(But, yes, this is a serious-and-important issue, in many ways, so, yeah, just some kidding-at-the-end.)