# Change rates of the 2nd Chebyshev function

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 $$h$$?

efq

There is the asymptotic estimate $\psi(x+h) - \psi(x) \sim h$ for $x^{7/12 + \epsilon} \leq h \leq x$, valid for any $\epsilon > 0$. This is due to M. N. Huxley, and dates to 1972. I am not aware of any better range for $h$ if you want asymptotic equality. But if you are satisfied with an order of magnitude result, you can have $c_1h \leq \psi(x+h) - \psi(x) \leq c_2h$ with $c_1$ and $c_2$ positive constants and $x^{\theta} \leq h \leq x$ for some $\theta$ slightly larger than $0.5$. I can't give references offhand, but you should be able to find such papers by searching on R. C. Baker, G. Harman, J. Pintz in Mathscinet.

• Thanks a lot for you answer! I believe this order-of-magnitude-result may in fact work out for me. I´ll definitely have a look into the reference you mention.
– M.G.
Nov 20, 2009 at 19:09

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.

• That is interesting. Is it known whether this null set is finite?
– M.G.
Nov 20, 2009 at 20:49
• there is a sequence of x_k -> oo such that psi(x_k + (log x_k)^2) - psi(x_k) > (1+epsilon) (log x_k)^2 ... (I hope this answers your question).
– maki
Nov 20, 2009 at 23:16
• yes, of course! ignore my last comment.
– M.G.
Nov 21, 2009 at 23:19