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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

flag	asked May 3 2010 at 21:35 Scott Guthery 639●3●6

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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/… – muad May 3 2010 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 2010 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 2010 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 2010 at 0:35

It is this equivalence that Franel uses in connecting Farey and Riemann. – Scott Guthery May 4 2010 at 10:34

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François G. Dorais · Accepted Answer · 2010-05-04 01:41:36Z UTC

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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.

answered May 4 2010 at 1:41

François G. Dorais♦
22.3k●2●48●113

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. – Marc Palm Nov 25 2010 at 15:42

Gerry Myerson · Answer 2 · 2010-05-04 13:00:43Z UTC

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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.

answered May 4 2010 at 13:00

Gerry Myerson
15.6k●1●47●87

Not quite. What I'd like is a citation for the equivalence that Franel uses in "Les suites...". – Scott Guthery May 4 2010 at 14:12

Jakob Katz · Answer 3 · 2010-05-04 19:27:54Z UTC

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.

$\zeta(s+1)/\zeta(s)$

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