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



$\zeta(s+1)/\zeta(s)$
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](http://en.wikipedia.org/wiki/M%C3%B6bius_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](http://en.wikipedia.org/wiki/Dirichlet_convolution).


