It is commonplace (see for instance this paper by McGown) to calculate the Riemann $ζ$ function at even integer values by using
- an approximation to the real value
- and the Staudt--von Clausen theorem.
My question is this: For the first item, why do we use the Euler product, instead of the definition of the Riemann $ζ$ function? The latter is, after all, a series. I would guess that the Euler product has a better convergence rate, but I don't see the proof just now.