Are the primes normally distributed? Or is this the Riemann hypothesis?

Question

Forgive my very naive question. I know next to nothing about number theory, but I'm curious about the state of the art on the distribution of primes.

Let $\mathrm{Li}(x)$ be the offset logarithmic integral, let $\pi(x)$ be the prime counting function, and recall that $$ \pi(x) \sim \mathrm{Li}(x). $$ For any $a,n\in\mathbb{N}$, let $$ I_n(a) \;=\; \bigl[a ,\ \mathrm{Li}^{-1}\bigl(\mathrm{Li}(a)+n\bigr)\bigr] $$ That is, $I_n(a)$ is the interval starting at $a$ in which we expect to find $n$ primes. Let $N_n(a)$ denote the actual number of primes in the interval $I_n(a)$, which defines a function $N_n\colon \mathbb{N}\to\mathbb{N}$.

Now, consider the following statement.

For all $n,k\in\mathbb{N}$, $$ \lim_{M\to\infty} \frac{\bigl|N_n^{-1}(\{k\}) \cap \{1,\ldots,M\}\bigr|}{M} \;=\; \frac{n^k e^{-n}}{k!}.\tag{1} $$

The quantity on the left is roughy the probability that, instead of finding $n$ primes in the interval $I_n(a)$, we find exactly $k$. The formula on the right is a Poisson distribution with parameter $n$. It is the naive thing one would expect this limit to converge to, assuming each number $m$ is prime with probability $1/\log(m)$.

Applying something like the central limit theorem to the above approximation, one obtains the statement

For all $x\in\mathbb{R}$, $$ \lim_{n\to\infty} \lim_{M\to\infty} \frac{\bigl|N_n^{-1}([0,n+x\sqrt{n}]) \cap \{1,\ldots,M\}\bigr|}{M} \;=\; \frac{1}{\sqrt{2\pi}} \int_{-\infty}^x e^{-t^2/2}dt\tag{2} $$

where the formula on the right is the cumulative distribution function of a standard normal distribution.

My question is:

What is the status of statements (1) and (2)?

In particular, are they (a) known to be true, (b) open but weaker than the Riemann hypothesis, (c) equivalent to the Riemann hypothesis, (d) stronger than the Riemann hypothesis, (e) unrelated to the Riemann hypothesis, or (f) known to be false?

Answer 1

These questions on the spacings between primes are expected to be true, but are far from being proved. They are not directly related to RH, but seem to encode other relations among zeros. The conjecture (1) follows from the Hardy-Littlewood prime $k$-tuples conjectures; this was established by Gallagher. More precise versions of Conjecture 2 were considered and formulated by Montgomery and Soundararajan. Here are a few references that address such questions:

1. Soundararajan. The distribution of prime numbers. http://arxiv.org/abs/math/0606408

2. Soundararajan. Small gaps between primes: the work of Goldston, Pintz and Yildirim. http://arxiv.org/abs/math/0605696 Now of course out of date(!), but does contain a discussion of spacings between primes.

3. Montgomery and Soundararajan: Primes in short intervals. http://arxiv.org/abs/math/0409258 Here you'll find versions of conjecture 2, and connections with zeros (but nothing as straightforward as direct link to RH).

Added: It may be helpful to give a brief description of the expected results. First, let us reformulate the problem as asking for an understanding of the number of primes in a random interval $[N,N+h]$ (with $h\le N$ say). For example, question (1) is concerned with the case $h=n\log N$ and the number of times such an interval contains $k$ primes. There are three natural ranges:

(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 $\lambda$ (as in (1) above). 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 $N$. In this context, the variance was first considered by Goldston and Montgomery, who found connections between this and the pair correlation conjectures for the zeros of $\zeta(s)$. The refined Gaussian conjectures may be thought of as saying that $X^{i\gamma}$ behave like independent random variables if $0\le \gamma \le T$ are the ordinates of zeros of $\zeta(s)$, and $X\ge T^{1+\delta}$. All this is in some sense beyond RH.

(3) The range when $h$ is a constant times $N$. This is like understanding the distribution of $\pi(x)-Li(x)$. After dividing by $\sqrt{x}/\log x$ one expects (on a logarithmic scale) a non-universal distribution here coming from the assumption that RH together with the linear independence of ordinates of zeros of $\zeta(s)$. Rubinstein and Sarnak's paper Chebyshev's Bias gives a good account of the flavor of such results.