By Bruce E. Larock, Roland W. Jeppson, Gary Z. Watters

Number concept, spectral geometry, and fractal geometry are interlinked during this in-depth learn of the vibrations of fractal strings, that's, one-dimensional drums with fractal boundary.

Key beneficial properties:

- The Riemann speculation is given a common geometric reformulation within the context of vibrating fractal strings

- complicated dimensions of a fractal string, outlined because the poles of an linked zeta functionality, are studied intimately, then used to appreciate the oscillations intrinsic to the corresponding fractal geometries and frequency spectra

- particular formulation are prolonged to use to the geometric, spectral, and dynamic zeta features linked to a fractal

- Examples of such formulation contain top Orbit Theorem with blunders time period for self-similar flows, and a tube formula

- the tactic of diophantine approximation is used to check self-similar strings and flows

- Analytical and geometric tools are used to acquire new effects concerning the vertical distribution of zeros of number-theoretic and different zeta functions

Throughout new effects are tested. the ultimate bankruptcy supplies a brand new definition of fractality because the presence of nonreal advanced dimensions with confident genuine parts.

The major reports and difficulties illuminated during this paintings can be utilized in a school room environment on the graduate point. Fractal Geometry, advanced Dimensions and Zeta services will entice scholars and researchers in quantity conception, fractal geometry, dynamical platforms, spectral geometry, and mathematical physics.

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**Additional resources for Fractal Geometry, Complex Dimensions and Zeta Functions: Geometry and Spectra of Fractal Strings**

**Example text**

For the spectral counting function, this follows from the following computation: ∞ ∞ Nν (x) = #{j : lj−1 ≤ x/k} = 1= k=1 j: k·l−1 ≤x j k=1 ∞ NL k=1 x . k Observe that this is a ﬁnite sum, since NL (y) = 0 for y < l1−1 . The second expression is derived similarly: ∞ ∞ Nν (x) = 1= j=1 k≤lj x [lj x]. j=1 For the spectral zeta function, we have successively ∞ ζν (s) = k,j=1 k −s ljs = ∞ ∞ ljs j=1 k −s = ζL (s)ζ(s). k=1 This completes the proof of the theorem. 37) for Nν (x), we can derive Weyl’s asymptotic law for fractal strings.

RN and gaps scaled by g1 , . . 4). 3. The gaps of L have lengths g1 L, . . , gK L. By abuse of language, we will usually refer to the quantities g1 , . . , gK as the gaps of the self-similar string L. 4. 1. Then the geometric zeta function of this string has a meromorphic continuation to the whole complex plane, given by ζL (s) = K s k=1 gk , N s j=1 rj Ls 1− for s ∈ C. 10) Here, L = ζL (1) is the total length of L, which is also the length of I, the initial interval from which L is constructed.

We note that by construction, the boundary ∂Ω of the Cantor string is equal to the ternary Cantor set. In general, the volume of the tubular neighborhood of the boundary of L is given by (see [LapPo2, Eq. 2), p. 48]) lj = 2ε · NL 2ε + V (ε) = j: lj ≥2ε j: lj <2ε 1 2ε lj . 3 When the two endpoints of an interval of length lj are covered by intervals of radius ε, then these discs overlap if lj < 2ε, covering a length lj , or they do not overlap if lj ≥ 2ε, in which case they cover a length of 2ε.