Big Oh of values of Riemann zeta function

Question

There is a equality in a proof in Apostol's Analytic Number Theory as follows: $O(x^{\alpha} \zeta(\alpha)) = O(x^{\alpha})$ for arbitrary real number $\alpha \ge 0$.

How do we say that? Does anything changes when $\alpha \in [0,1]$?

Answer 1

The thing is - as far as the implied limit there is concerned - for fixed $\alpha$ as $x$ varies, $\zeta(\alpha)$ is just a number. $O(cf(x))=O(f(x))$ for any constant $c$ and function $f$.

As noted in the comments, $\alpha=1$ is an exception; there, $O(x^{\alpha}\zeta(\alpha))$ becomes $O(\infty)$, which is vacuously true for any function of $x$.

In practice, the places you'll actually want to use this always come from values of $\alpha$ for which the series for $\zeta$ converges - namely, $\alpha>1$. However, it's still true for $\alpha<1$, using the definitionof $\zeta$ by analytic continuation.