There is an elementary statement that I *believe* I have read somewhere, but I can't remember where. I'd like to know if the statement is correct (in which case it is surely standard) and if so, where I can find a proof of it.

The statement is about the prime-counting function $\psi(x) = \sum_{ p^n < x } \log p$. It is well-known that GRH is equivalent to $\psi(x) = x + O(x^{1/2} \log^2 x)$. The statement in question is:

If $1/2< \alpha < 1$ is a real number, and $\zeta(s)$ has no zero on $\Re s> \alpha$, then $\psi(x) = x + O(x^{\alpha} \log x)$?

The point here is that we get to replace $\log^2 x$ by $\log x$. Thanks for any reference (or disproof, if this is just a dream I made).