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Let us consider the number of primes in the interval $[N,N+h]$, with $h\leq N$. According to the answer given by Lucia to a previous question on the distribution of primes, it is natural to consider the three ranges for $h$ (the following is an excerpt from that answer):

(1) $h = \lambda \log N$ for a fixed $\lambda$ (and $N$ chosen randomly). Here one expects the number of primes to be Poisson with parameter $\lambda$. This is what Gallagher showed follows from Hardy-Littlewood, and this is consistent with the Cramer model that primes are like random numbers thrown down with mean spacing $\log N$.

(2) $h/\log N \to \infty$ but $h/N \to 0$. Here one expects a normal distribution with mean $h/\log N$ and variance $h (\log N/h)/(\log N)^2$. This is the conjecture of Montgomery and Soundararajan, and is a little bit different from the Cramer model (which would predict a variance of $h/\log N$); the difference is noticeable when $h$ is a power of $N$.

(3) The range when $h$ is a constant times $N$. This is like understanding the distribution of $\pi(x)-Li(x)$.

In the case of (2), let us also look at the sub-ranges

i) $h=(log N)^\lambda$, $\lambda>1$, and

ii) $h=N^\delta$, $0<\delta<1$.

If we now let the mean number of primes in $[N,N+h]$ be denoted by $\mu_h$ and variance by $var_h$, we can observe that the relationship $\mu_h/var_h$ is linear in cases (1) and (2) above:

(1) $\mu_h/var_h \sim 1$,

(2 i) $\mu_h/var_h \sim 1$,

(2 ii) $\mu_h/var_h \sim 1/(1-\delta)$.

In the case of (3), it might be hypothesized that $\mu_h/var_h \rightarrow \infty$ as $N\rightarrow \infty$.

Maier's theorem says that if $h$ grows at most as fast as (2 i), we can always find a set of intervals in which the density differs with a multiplicative constant from the prime number theorem. In the context of Montgomery and Soundararajan's conjecture, my naive (and perhaps confused) interpretation of Maier's theorem is that since the ratio $\mu_h/var_h$ is linear, we can always pick a set of intervals that deviates from the mean by a constant, as long as this constant is within the finite borders of the distribution. However, this argument (if it came through clearly) should translate also to (2 ii). But both Granville, p. 7 and Soundararajan, p. 79 conjecture (if i read them correctly) that the prime number theorem should hold for all $N$ as long as $h\geq N^\delta$, i.e. the case (2 ii), which seems to contradict my interpretation of Maier's theorem.

So I guess my question then is:

Can Maier's theorem be understood intuitively from Montgomery and Soundararajan's conjecture? And if so, where do I run wild in my arguments to end up contradicting the conjecture by Granville and Soundararajan?

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Are you possibly confusing variance and standard deviation? E.g. if $h=(\log N)^2$ then the Montgomery-Soundararajan conjecture says that the number of primes is about $\log N$ with Gaussian fluctuations on the scale of $\sqrt{\log N}$ (which is the standard deviation). The Maier phenomenon is a rare large deviations event, which doesn't fit the tails of the Gaussian. –  Lucia May 21 at 17:03
    
It seems you're right about my confusion. I went about it too quickly I must admit. Anyhow, it got me thinking and your comment made a few pieces snap in place, which in the end was the point of asking. Thanks (again)! –  user45947 May 21 at 22:59
    
Glad that was useful! –  Lucia May 22 at 0:24

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