change rates of the 2nd Chebychev´s function - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T13:52:32Z http://mathoverflow.net/feeds/question/5761 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/5761/change-rates-of-the-2nd-chebychevs-function change rates of the 2nd Chebychev´s function ex falso quodlibet 2009-11-17T01:07:30Z 2009-11-19T05:39:45Z <p>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 $h$?</p> <p>Thanks in advance,</p> <p>efq</p> http://mathoverflow.net/questions/5761/change-rates-of-the-2nd-chebychevs-function/5836#5836 Answer by engelbrekt for change rates of the 2nd Chebychev´s function engelbrekt 2009-11-17T16:48:30Z 2009-11-17T22:18:03Z <p>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.</p> http://mathoverflow.net/questions/5761/change-rates-of-the-2nd-chebychevs-function/6064#6064 Answer by maki for change rates of the 2nd Chebychev´s function maki 2009-11-19T05:39:45Z 2009-11-19T05:39:45Z <p>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.</p>