An Introduction to the Theory of Numbers by Vinogradov I. M.

By Vinogradov I. M.

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On the other hand, if L has finitely many lengths, then D = 0 but σ = −∞ since the series for ζL (s) converges for all s ∈ C. 9. The abscissa of convergence of the series fined by σ = inf α ∈ R : ∞ j=1 ljα < ∞ . 20) Thus {s ∈ C : Re s > σ} is the largest open half-plane on which this series ∞ converges. 2]. The next theorem relates σ with D. 10. Suppose L has infinitely many lengths. Then the abscissa of convergence of the geometric zeta function of L coincides with D, the Minkowski dimension of ∂L.

Proof. 15 or from [Pos] or [Wid] that D is a singularity of ζL . Suppose NL (x) ≤ C · xD and NL (x) = 0 for x ≤ x0 . Then for s > D, ∞ ζL (s) = s NL (x)x−s−1 dx ≤ 0 Cs D−s x . s−D 0 It follows that the singularity at D is at most a simple pole. Since by assumption, ζL has a meromorphic extension to a neighborhood of D, it follows that D is a simple pole of ζL . 1 since V (ε) = 2εNL 1 + 2ε lj = O ε1−D , as ε → 0+ , j: lj <2ε implies that NL (x) = O xD as x → ∞. The following example shows that the condition that the geometric zeta function of a fractal string has a meromorphic continuation is not always satisfied, and indeed the vertical line Re s = D can be a natural boundary for the analytic continuation of ζL .

1, we propose as a new definition of fractality the presence of at least one nonreal complex dimension with positive real part. In this new sense, every self-similar set in the real line is fractal. On the other hand, in agreement with geometric intuition, certain compact subsets of R, associated with the so-called a-string, are shown here to be nonfractal, whereas they are fractal according to the definition of fractality based on the notion of Minkowski dimension. 6). In particular, the Devil’s staircase is not fractal according to the traditional definition based on the Hausdorff dimension.

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