By W.N. Bailey

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During this quantity, that is devoted to H. Seifert, are papers in line with talks given on the Isle of Thorns convention on low dimensional topology held in 1982.

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However, for simplicity, we do not indicate this explicitly in our notation. 18. 35) f which converges for Re s suﬃciently large. 35), the sum is extended over all distinct frequencies of L. 9 According to footnote 1, the frequency f = x should be counted with the weight 1/2. 24 1. Complex Dimensions of Ordinary Fractal Strings ∞ Let ζ(s) be the Riemann zeta function deﬁned by ζ(s) = n=1 n−s for Re s > 1. It is well known that ζ(s) has an extension to the whole complex plane as a meromorphic function, with one simple pole at s = 1, with residue 1.

13. 10, ζL (s) is holomorphic in the half-plane Re s > D and hence DL = DL (W ) ⊂ {s ∈ W : Re s ≤ D} . 26) Moreover, since it is the set of poles of a meromorphic function, DL (W ) is a discrete subset of C. Hence its intersection with any compact subset of C is ﬁnite. When L consists of ﬁnitely many lengths, we have DL = ∅, since then, ζL (s) is an entire function. 14. , from [Ser, Proposition 7, p. 18) has positive coeﬃcients). Thus s = DL is always a singularity of ζL (s), but not necessarily a pole.

9). 1 The Multiplicity of the Lengths Another way of representing a fractal string L is by listing its diﬀerent lengths l, together with their multiplicity wl : wl = #{j ≥ 1 : lj = l}. 7) Thus, for example, NL (x) = wl . 8) l−1 ≤x In Chapter 4, we will introduce a third way to represent a fractal string, similar to this one, namely, by a measure. 2: The Cantor string. 037-tubular neighborhood of the Cantor string. 2). Thus CS = ( 13 , 23 )∪( 19 , 29 )∪( 79 , 89 )∪ 1 2 27 , 27 ∪ 7 8 27 , 27 ∪ 19 20 27 , 27 ∪ 25 26 27 , 27 ∪ s, ˙ so that l1 = 1/3, l2 = l3 = 1/9, l4 = l5 = l6 = l7 = 1/27, .