Find a Dirichlet series for $\frac{\zeta(s-1)}{\zeta(s)}$ valid for $Re(s)>2$.

Question

Find a Dirichlet series for $\frac{\zeta(s-1)}{\zeta(s)}$ valid for $Re(s)>2$.

I know that we should use absolute convergence but not sure how that applies in this case.

Answer 1

If the Dirichlet series of $f$ and $g$ converge absolutely for some $s$, then $\text{D}(f,s)\cdot \text{D}(g,s)=\text{D}(f*g,s)$.

We know that $\text{D}(\mu,s)=\frac1{\zeta(s)}$ so it suffices to find a function whose dirichlet series is $\zeta(s-1)$; one easily sees that it is the function $N(n)=n$.

Hence $\frac{\zeta(s-1)}{\zeta(s)}=\text{D}(N*\mu,s)$ for $\text{Re}(s)>2$.

Now we use Möbius inversion to find out what $N*\mu$ is. Since $N*\mu=f$ iff $N=f*u$, where $u$ is the constant $1$ function, we seek for an $f$ which satisfy $\sum\limits_{d|n}f(d)=n$. This is satisfied by Euler's totient function $\varphi$, so we get that $\frac{\zeta(s-1)}{\zeta(s)}=\text{D}(\varphi,s)$.