TITLE: Change rates of the 2nd Chebyshev function
QUESTION [6 upvotes]: Let $\psi(x):=\sum_{n\leq x}\Lambda(n)$ denotes the 2nd Chebyshev function, where $\Lambda$ stands for the von Mangoldt function. Are there any known (and 'nice') estimates for the change rates $\psi(x+h)-\psi(x)$ for general or special $x$ and $h$?
Thanks in advance,
efq

REPLY [2 votes]: Also, I think Selberg's result asserts that for "almost all x"
we have $\psi(x + h) - \psi(x) \sim h$ for $h \asymp (\log x)^2$; 
this was shown to not be true "pointwise" by Mayer.