$\zeta(s+1)/\zeta(s)$ - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-18T19:44:43Z http://mathoverflow.net/feeds/question/23378 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/23378/zetas1-zetas $\zeta(s+1)/\zeta(s)$ Scott Guthery 2010-05-03T21:35:35Z 2010-05-04T19:27:54Z <p>Franel uses the convergence of</p> <p>$\frac{\zeta(s+1)}{\zeta(s)} = \sum \frac{c(n)}{n^s}$</p> <p>as an equivalent to the Riemann hypothesis.</p> <p>Does anybody have a citation for this result and/or hints for computing $c(n)$?</p> <p>Thanks for any insight.</p> <p>Cheers, Scott</p> http://mathoverflow.net/questions/23378/zetas1-zetas/23395#23395 Answer by François G. Dorais for $\zeta(s+1)/\zeta(s)$ François G. Dorais 2010-05-04T01:41:36Z 2010-05-04T01:41:36Z <p>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 <a href="http://en.wikipedia.org/wiki/M%C3%B6bius_function" rel="nofollow">Möbius function</a>, we have $$c(n) = \sum_{d \mid n} \frac{d}{n}\mu(d) = \frac{1}{n}\prod_{p \mid n} (1-p)$$ using <a href="http://en.wikipedia.org/wiki/Dirichlet_convolution" rel="nofollow">Dirichlet convolution</a>.</p> http://mathoverflow.net/questions/23378/zetas1-zetas/23421#23421 Answer by Gerry Myerson for $\zeta(s+1)/\zeta(s)$ Gerry Myerson 2010-05-04T13:00:43Z 2010-05-04T13:00:43Z <p>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. </p> http://mathoverflow.net/questions/23378/zetas1-zetas/23470#23470 Answer by Jakob Katz for $\zeta(s+1)/\zeta(s)$ Jakob Katz 2010-05-04T19:27:54Z 2010-05-04T19:27:54Z <p>This would be a comment, but I don't have the points for it.</p> <p>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.</p>