By Ioannis Dimitrios Avgoustis

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1, and the notes to Chapter 9. The primary motivations of the authors of those papers came from the investigation of the oscillatory phenomena in the geometry and the spectrum of fractal drums [Lap3] (including self-similar drums) and, in particular, of fractal strings, where the connections between direct or inverse spectral problems and the Riemann zeta function or the Riemann hypothesis were ﬁrst discovered in [LapPo1, 2] or [LapMa1, 2], respectively. 17 for a sample of references in the physics literature—of which we have become aware recently and with rather diﬀerent motivations, coming from the study of turbulence, lacunarity, biophysics, and other applications.

We note that such strings can no longer be realized geometrically as subsets of Euclidean space. 4 Higher-Dimensional Analogue: Fractal Sprays Fractal sprays were introduced in [LapPo3] (see also [Lap2, §4] announcing some of the results in [LapPo3]) as a natural higher-dimensional analogue of fractal strings and as a tool to explore various conjectures about the spectrum (and the geometry) of drums with fractal boundary10 in Rd . In the present book, fractal sprays and their generalizations (to be introduced later on) will continue to be a useful exploratory tool and will enable us to extend several of our results to zeta functions other than the Riemann zeta function.

17) The Geometric Zeta Function of a Fractal String ∞ ∞ Let L be a fractal string with sequence of lengths {lj }j=1 . The sum j=1 ljσ converges for σ = 1. It follows that the (generalized) Dirichlet series ∞ ljs ζL (s) = j=1 deﬁnes a holomorphic function for Re s > 1. 10 below that this series converges in the open right half-plane Re s > D, deﬁned by the Minkowski dimension D, but that it diverges at s = D. 7) above for the deﬁnition of the multiplicity wl in the following deﬁnition. 6 Note that since p = 2π/ log 3 and u = − log ε, we have eipu = ε−ip .