Wikipedia (here) says that $\frac{\zeta(s-1)}{\zeta(s)}= \sum_{n=1}^{\infty}\frac{\varphi(n)}{n^{s}}$ where $\varphi(n)$ is the totient function. Similarly, is there a known expression involving a Dirichlet series for $\frac{\zeta(1-s)}{\zeta(s)}$?
Thanks, Gregory
$\zeta(1-s)$ is not a Dirichlet series (which by definition is $f(s)=\sum a_n n^{-s}$) since for example, it doesn't have a (finite) limit as $\Re s \to \infty$ which all Dirichlet series that converge somewhere do, so it doesn't make sense to ask about a Dirichlet series as above.
By the functional equation, we know that $$\frac{\zeta(1-s)}{\zeta(s)}=2^{1-s}\pi^{-s}\cos \frac{s\pi}{2}\Gamma(s)$$ so we know an expression for it but again the function (meromorphic with poles at $s=-2n$) is not a Dirichlet series
The symmetry of $\zeta$ is $s \to 1-s$ not $s \to -s$ so $\zeta(1-s)$ and $\zeta(s-1)$ do not have a priori any particular relationship