By W.N. Bailey

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Low Dimensional Topology

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, .