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A fractal drum is a bounded open subset of ℝ with a fractal boundary. A difficult problem is to describe the relationship between the shape (geometry) of the drum and its sound (its spectrum). In this book, we restrict ourselves to the one-dimensional case of fractal strings, and their higher-dimensional analogues, fractal sprays. We develop a theory of complex dimensions of fractal strings, and we study how these complex dimensions relate the geometry and the spectrum of fractal strings. See the notes to Chapter 1 in Section 1.5 for references to the literature.

In this chapter, we recall some basic definitions pertaining to the notion of (ordinary) fractal string and introduce several new ones, the most important of which is the notion of complex dimension. We also give a brief overview of some of our results in this context by discussing the simple but illustrative example of the Cantor string. In the last section, we discuss fractal sprays, which are a higher-dimensional analogue of fractal strings.

Throughout this book, we use an important class of ordinary fractal strings, the self-similar fractal strings, to illustrate our theory. These strings are constructed in the usual way via contraction mappings. In this and the next chapter, we give a detailed analysis of the structure of the complex dimensions of such fractal strings.

The study of the complex dimensions of nonlattice self-similar strings is most naturally carried out in the more general setting of Dirichlet polynomials. In this chapter, we study the solutions in of a

In this chapter, we develop the notion of generalized fractal string, viewed as a measure on the half-line. This is more general than the notion of fractal string considered in Chapter 1 and in the earlier work on this subject (see the notes to Chapter 1). We will use this notion in Chapter 5 to formulate the explicit formulas which will be applied throughout the remaining chapters. Besides ordinary fractal strings, generalized fractal strings enable us to deal with strings whose lengths vary continuously or whose multiplicities are nonintegral or even infinitesimal. In Section 4.2, we discuss the spectrum of a generalized fractal string, and in Section 4.3, we briefly discuss the notion of generalized fractal spray, which will be used in Chapters 9 and 11.

In this chapter, we obtain pointwise and distributional explicit formulas for the lengths and the frequencies of a fractal string. These explicit formulas express the counting function of the lengths or of the frequencies as a sum over the visible complex dimensions of the fractal string. To unify the exposition, and with a view toward later applications, we formulate our results in the language of generalized fractal strings, introduced in Chapter 4.

In this chapter, we give various examples of explicit formulas for the counting function of the lengths and frequencies of (generalized) fractal strings and sprays.

In this chapter, we apply our explicit formulas to obtain an asymptotic expansion for the prime orbit counting function of suspended flows. The resulting formula involves a sum of oscillatory terms associated with the dynamical complex dimensions of the flow. We then focus in Section 7.3 on the special case of self-similar flows and deduce from our explicit formulas a Prime Orbit Theorem with error term. For a self-similar flow, we define the lattice and the nonlattice case in Definition 7.27. In the lattice case, the counting function of the prime orbits has oscillatory leading asymptotics. The explicit formula for this counting function enables us to give a very precise expression for this function in terms of multiplicatively periodic functions. In the nonlattice case (which is the generic case), the leading term does not have oscillations, and we provide a detailed analysis of the error term. The precise order of the error term depends on the dimension-free region of the dynamical zeta function, as in the classical Prime Number Theorem. Applying the results of Chapter 3, we find that this region, and hence the error term, depends on properties of Diophantine approximation of the weights of the flow.

In this chapter, we obtain (in Section 8.1) a distributional formula for the volume of the tubular neighborhoods of the boundary of a fractal string, called a . In Section 8.1.1, under more restrictive assumptions, we also derive a tube formula that holds pointwise. In Section 8.3, we then deduce from these formulas a new criterion for the Minkowski measurability of a fractal string, in terms of its complex dimensions. Namely, under suitable assumptions, we show that a fractal string is Minkowski measurable if and only if it does not have any nonreal complex dimensions of real part , its Minkowski dimension. This completes and extends the earlier criterion obtained in [LapPo1-2].

In this chapter, we provide an alternative formulation of the Riemann hypothesis in terms of a natural inverse spectral problem for fractal strings. After stating this inverse problem in Section 9.1, we show in Section 9.2 that its solution is equivalent to the nonexistence of critical zeros of the Riemann zeta function on a given vertical line. This modifies and extends the earlier work of [LapMa1-2], but now we use the point of view of complex dimensions and the explicit formulas of Chapter 5. In Section 9.3, we then extend this characterization to a large class of zeta functions, including all the number-theoretic zeta functions for which the extended Riemann hypothesis is expected to hold.