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I have been told that, for a given sequence ${N_r}$, its zeta-function is given by $exp(\Sigma _{r=1}^{\infty} N_rT^r/r)$. Since I have barely any experience with such a sum, I tried to find some familiar examples to start with. But I failed to tell what the zeta-function for the Fibonacci sequence is.
WHat I have tried so far is: By use of the closed-form, we might tell that $N_r=\frac{(1 + \sqrt{5})^r - (1 - \sqrt{5})^r}{2^r \sqrt{5}}$. Hence, by means of logarithms, we can derive that the function ought to be equal to ${(1-(1 - \sqrt{5})T/2)/[(1-(1 + \sqrt{5})T/2)]}^{1/ \sqrt{5}}$. But this formula seems to be just to... hard to accept. So I suspect that I must be wrong somewhere. Then am I wrong? And are there other proofs of this relation? And, in general, how to derive the zeta-function for a given sequence? Thanks in advance.

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It looks right. Can you give a reference on zeta function on a sequence in general? –  user27126 Feb 1 '13 at 15:39
    
@Sanchez I read about this topic in this book, an interesting and inspiring book on algebraic geometry via analytic ways. However it is not on zeta-functions of sequences in general, insofaras I know. Moreover, it is on the second chapter. –  awllower Feb 1 '13 at 15:42
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There is a reason to consider zeta function for the sequence "number of points of a variety mod p^n". I'm not sure if the zeta function for a general sequence is of interest. –  user27126 Feb 1 '13 at 15:52
    
@Sanchez Indeed. There is yet no indication of their usefulness. But, in connection with the generating function of a sequence, it could not be deemed entirely useless however. Also, I agree that the study of zeta-function of #V$(E/F_{q^r})$ leads to some deep understandings of the arithmetic properties of an algebraic variety. –  awllower Feb 1 '13 at 15:54

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