Sum of totient equals zeta(z-1)/zeta(z) for Re(z) greater than 2

Theorem

The following formula holds for $\mathrm{Re}(z) > 2$: $$\displaystyle\sum_{k=1}^{\infty} \dfrac{\phi(k)}{k^z} = \dfrac{\zeta(z-1)}{\zeta(z)},$$ where $\phi$ denotes the totient and $\zeta$ denotes the Riemann zeta function.

References

• 1964: Milton Abramowitz and Irene A. Stegun: Handbook of mathematical functions ... (previous) ... (next): $24.3.2 \mathrm{I}.B.$