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Spectral zeta functions. (English) Zbl 0819.11033
Kurokawa, N. (ed.) et al., Zeta functions in geometry. Tokyo: Kinokuniya Company Ltd.. Adv. Stud. Pure Math. 21, 327-358 (1992).
This illuminating survey paper discusses zeta functions for spectral data. It treats the relations between the zeta function $$\sum \rho_ k^{-s}$$, the theta function $$\sum e^{-t \rho_ k}$$, and the regularized determinant $$\prod (\rho_ k+ \rho)$$. The quantities $$\rho_ 0, \rho_ 1, \rho_ 2, \dots$$ may be the eigenvalues $$\lambda_ k$$, or a distortion $$\rho_ k= \rho (\lambda_ k)$$. As the basis example the author discusses the zeta function of Riemann. Next he treats the spectrum of the Laplacian on compact hyperbolic surfaces. Here the trace formula of Selberg is an important tool. The last example is the spectrum of the operator $$x^ N- ({\partial \over {\partial x}})^ 2$$ in $$L^ 2 (\mathbb{R})$$, with even $$N\geq 4$$. Investigation of a theta function associated to this spectrum leads to a resurgence algebra.
For the entire collection see [Zbl 0771.00036].

##### MSC:
 11M41 Other Dirichlet series and zeta functions 58J50 Spectral problems; spectral geometry; scattering theory on manifolds 11-02 Research exposition (monographs, survey articles) pertaining to number theory 11F72 Spectral theory; trace formulas (e.g., that of Selberg) 11M06 $$\zeta (s)$$ and $$L(s, \chi)$$ 34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)