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Generalized Minkowski content, spectrum of fractal strings, and the Riemann zeta-function. (English) Zbl 0877.35086
Mem. Am. Math. Soc. 608, 97 p. (1997).
This memoir is a survey on the effect of the fractality of the domain on the spectrum of the Dirichlet or Neumann Laplace operator. The authors study especially fractal strings and the links between the eigenvalues of the Laplace operator, the counting function, the Minkowski contents, and the Riemann zeta-function.

MSC:
35P20 Asymptotic distributions of eigenvalues in context of PDEs
11M06 \(\zeta (s)\) and \(L(s, \chi)\)
28A80 Fractals
34B24 Sturm-Liouville theory
37A30 Ergodic theorems, spectral theory, Markov operators
58J50 Spectral problems; spectral geometry; scattering theory on manifolds
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