Complex Geometry Consequences of Serre's Kähler-Zeta Function Serre's famous paper Analogues Kählériens de Certaines Conjectures de Weil proves an analogue of the Weil conjectures for compact Kähler manifolds. It would go on to inspire the line of attack that eventually solved the Weil conjectures. 
I would like to know is whether the result had any applications or consequences in complex geometry. Since I guess this is a little broad of a question for this site, let me write: Can anyone name a substantial result in complex geometry that uses Serre's zeta function in a non-trivial way. (Apologies for fluffy words like substantial and non-trivial.)
 A: This is a purity result for a polarized Kähler dynamical system. The precise statement is that if $(X,\omega)$ is compact Kähler and $\phi : X \to X$ an endomorphism having the class $[\omega] \in H^2(X,\mathbb{R})$ as eigenvector of eigenvalue $q > 1$ (a polarized Kähler dynamical system of positive entropy), then all eigenvalues for the action of $\phi^*$ on $H^i(X,\mathbb{R})$ have modulus $\sqrt{q}^i$.
A number of consequences of this result for algebraic and Kähler dynamics are derived by Shou-wu Zhang in [Distributions in algebraic dynamics, Surveys in Differential Geometry, vol. X, 2006], which also reproduces the proof of Serre's theorem in mildly generalized form.
As for the zeta function, not explicitly considered in Serre's paper, it is a special case of a dynamical zeta, which is a vast subject. 
Incidentally, there is something else going by the name of Kähler zeta function, discussed in Anton Deitmar's article A panorama of zeta functions from the volume [Erich Kähler. Mathematische Werke., De Gruyter, 2003], also available on ArXiv. Kähler's zeta function is actually arithmetical; it is a different idea of generalizing Riemann's zeta to finitely generated fields than the much better known (and better behaved) Hasse-Weil zeta function. Kähler's interest in finitely generated fields is well known (suffice it to recall the notion of the Kähler differential), but I was amused to learn from J.-B. Bost's article in that volume (A neglected aspect of Kähler's work in arithmetic geometry) that the modern subject of Arithmetic Geometry apparently takes its name from the title of a paper (Geometria Aritmetica) that Kähler wrote in Italian, and that Kähler's writings contain the germs of ideas and notions that later found their proper place in Arakelov theory, such as the Faltings height of an abelian variety.
