change rates of the 2nd Chebychev´s function - MathOverflow

change rates of the 2nd Chebychev´s function

2009-11-17T01:07:30Z

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

Answer by engelbrekt for change rates of the 2nd Chebychev´s function

2009-11-17T16:48:30Z

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.

Answer by maki for change rates of the 2nd Chebychev´s function

2009-11-19T05:39:45Z

Also, I think Selberg's result assert 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.