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Let the zeta function of a fractal string be $ Z(s)= \sum_{n}g(n)(l_{n})^{s} $

Here $ g(n) $4 is the degeneracy of the strings and $ l_{n} $ are the lengths of the string.

In order to evaluate the dimension of the fractal string we must solve and find the poles of the zeta function $ Z(s)= \infty $ (why is this ?).

However, my question is a bit different, let us suppose we know all (or at least we can evaluate them) only from the length of the strings can we evaluate the dimension of the fractal string?

For example, in the Cantor case $ l_{n}=3^{-n} $ and $ g(n)=2 $ without solving any equations and only from the strings could we get the dimension of the Cantor set? Thanks.

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Why is one question mark per question not enough? –  joriki Jun 13 '12 at 15:49
SO you mean this? Find the poles of $Z(s)=\sum_{n=0}^\infty 2\cdot 3^{-ns}$ without solving any equation...??? Even though it is a geometric series, and we have a formula for it, still finding the poles involves "solving an equation", namely finding when the denominator is zero. –  GEdgar Jun 13 '12 at 15:55
Another comment. Merely to find the "dimension" we don't need all the poles, but just the maximum of the real parts of the poles. And (in the Cantor case) we can do that with the ratio test. –  GEdgar Jun 13 '12 at 15:57

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