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The Riemann zeta-function and the one-dimensional Weyl-Berry conjecture for fractal drums. (English) Zbl 0739.34065
Based on his earlier work on the vibrations of “drums with fractal boundary”, the first author has refined M. V. Berry’s conjecture that extended from the “smooth” to the “fractal” case H. Weyl’s conjecture for the asymptotics of the eigenvalues of the Laplacian on a bounded open subsetof \(\mathbb{R}^ n\). (See M. L. Lapidus, “Fractal drum, inverse spectral problems for elliptic operators and a partial resolution of the Weyl-Berry conjecture,” Trans. Am. Math. Soc., in press.) We solve here in the one-dimensional case (i.e., when \(n=1\)) this “modified Weyl-Berry conjecture”. We discover, in the process, some unexpected and intriguing connections between spectral geometry, fractal geometry and the Riemann zeta-function. We therefore show that one can not only “hear” (i.e., recover from the spectrum) the Minkowski fractal dimension of the boundary — as was established previously by the first author — but also, under the stronger assumptions of the conjecture, its Minkowski content (a “fractal” analogue of its “length”). We also prove (still in dimension one) a related conjecture of the first author, as well as its converse, which characterizes the situation when the error estimates of the aforementioned paper are sharp.
Reviewer: M.L.Lapidus

MSC:
34L20 Asymptotic distribution of eigenvalues, asymptotic theory of eigenfunctions for ordinary differential operators
34B20 Weyl theory and its generalizations for ordinary differential equations
28A12 Contents, measures, outer measures, capacities
11M06 \(\zeta (s)\) and \(L(s, \chi)\)
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