I have read this article (Montgomery and Soundararajan: Primes in short intervals. http://arxiv.org/abs/math/0409258 ). In the second page of the article, it is stated that the mean and variance of $\psi(n+h)-\psi(n)$, in a given range of h and n, tends to $h$ and $h \log\frac{h}{n}$ respectively, being $\psi(n)$ the second Chebyshev function. On the other hand, I have read this question (Are the primes normally distributed? Or is this the Riemann hypothesis?) in which it is stated that the mean and variance of $\pi(n+h)-\pi(n)$ tends to $\frac{h}{\log n}$ and $\frac{h}{(\log n)^{2}} \log\frac{h}{n}$ in that range. My question is, ¿ How can I obtain the moments of the prime counting function from the moments of the second Chebyshev function. ?
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1$\begingroup$ Should be $\log n/h$ throughout. In a short interval $[n,n+h]$ the weighting by $\Lambda$ is just weighting by approximately $\log n$; don't worry about prime powers since there are so few of those. $\endgroup$– LuciaMar 22, 2020 at 17:51
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