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).



Estimate on the prime-counting function $\psi(x)$.
The reference for this (given in the 3rd edition of Davenport's *Multiplicative Number Theory*) is:


Grosswald, Émile. "Sur l'ordre de grandeur des différences $\psi(x)-x$ et $\pi(x)-\ell i(x)$." (French) 
C. R. Acad. Sci. Paris **260** (1965), 3813–3816.


It seems (according to the Math Review) that for $\alpha$ fixed, the error is $O(x^\alpha)$. So, two powers of the logarithm can be saved.


