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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.