Contemporary MathematicsComplex Dimensions of Fractal Strings
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چکیده
We put the theory of Dirichlet series and integrals in the geometric setting of`fractal strings' (one-dimensional drums with fractal boundary). The poles of a Dirichlet series thus acquire the geometric meaning of`complex dimensions' of the associated fractal string, and they describe the geometric and spectral oscillations of this string by means of anèxplicit formula'. We deene thèspectral operator', which allows us to characterize the presence of critical zeros of zeta-functions from a large class of Dirichlet series as the question of invertibility of this operator. We thus obtain a geometric reformulation of the generalized Riemann Hypothesis, thereby extending the earlier work of the rst author with H. Maier. By considering the restriction of this operator to the subclass of`generalized Cantor strings', we prove that zeta-functions from a large subclass of this class have no innnite sequence of zeros forming a vertical arithmetic progression. (For the special case of the Riemann zeta-function, this is Putnam's theorem.) We make an extensive study of the complex dimensions of`self-similar' fractal strings, to gain further insight into the kind of geometric information contained in the complex dimensions. We also obtain a formula for the volume of the tubular neighborhoods of a fractal string and draw an analogy with Riemannian geometry. Our work suggests to deenèfractality' as the presence of nonreal complex dimensions with positive real part. c 0000 (copyright holder) 1 2 MICHEL L. LAPIDUS AND MACHIEL VAN FRANKENHUYSEN
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تاریخ انتشار 2009