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Franel uses the convergence of

$ \frac{\zeta(s+1)}{\zeta(s)} = \sum \frac{c(n)}{n^s}$

as an equivalent to the Riemann hypothesis.

Does anybody have a citation for this result and/or hints for computing $c(n)$?

Thanks for any insight.

Cheers, Scott

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This is not the same series as you have mentioned but it may be useful none-the-less. Warren D. Smith, "Cruel and unusual behavior of the Riemann zeta function" secamlocal.ex.ac.uk/people/staff/mrwatkin/zeta/cruel.ps –  muad May 3 '10 at 21:49
    
To compute $c(n)$ use the identities $1/zeta(s) = \sum_{n=1}^{\infty} \mu(n)/n^{s}$ (this follows from Mobius inversion) and $\zeta(s+1) = \sum_{n=1}^{\infty} 1/n^{s+1}$, which hold in an appropriate half-plane. –  Mark Lewko May 3 '10 at 21:57
    
The computation is discussed in G. P\'olya and G.~Szeg\"o, Problems and theorems in analysis, Vol.~II, Grundlehren Math. Wiss. 216, Springer-Verlag, Berlin et al. (1976), Division 8, Chapter 1, Sections 5--7. –  Wadim Zudilin May 3 '10 at 22:14
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The only Franel-Riemann connection I've been able to find concerns Farey series. How have you come to believe that Franel did what you say he did? –  Gerry Myerson May 4 '10 at 0:35
    
It is this equivalence that Franel uses in connecting Farey and Riemann. –  Scott Guthery May 4 '10 at 10:34
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3 Answers 3

up vote 9 down vote accepted

Since $$\zeta(s+1) = \sum_{n=1}^\infty \frac{1/n}{n^s}$$ and $$\frac{1}{\zeta(s)} = \sum_{n=1}^\infty \frac{\mu(n)}{n^s}$$ where $\mu$ is the Möbius function, we have $$c(n) = \sum_{d \mid n} \frac{d}{n}\mu(d) = \frac{1}{n}\prod_{p \mid n} (1-p)$$ using Dirichlet convolution.

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The conditional convergence of $\sum_{n=1}^{\infty} \mu(n) n^{-s}$ in $\Re s >1/2$ is equivalent to the Riemann Hypothesis. We know that $\sum_{n=1}^{\infty} n^{-s-1}$ converges for $\Re s >0$. From these two fact, you can deduce that the convergence of $\zeta(s+1)/\zeta(s)$ in $\Re s >1/2$ is actually equivalent to RH. –  plusepsilon.de Nov 25 '10 at 15:42
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If I understand correctly, what Scott wants is a citation for Franel's paper on (Farey series and) the Riemann Hypothesis. That would be Les suites de Farey et le problème des nombres premiers, Göttinger Nachr. (1924) 198–201.

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Not quite. What I'd like is a citation for the equivalence that Franel uses in "Les suites...". –  Scott Guthery May 4 '10 at 14:12
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This would be a comment, but I don't have the points for it.

Also not quite what you're asking for, but for a different look at what I believe is the same relationship between Farey series and RH (not having read Franel's paper), check H.M. Edwards' book on the Riemann zeta function, paragraph 12.2. He references Franel, but proves the equivalence in question by other means.

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