Let π(x)=#{p≤x∣p is prime} be the prime counting function. The Prime Number Theorem tells us that
π(x)∼xlogx.
(That is
limx→∞π(x)x/logx=1.) So, roughly speaking, around a large
x, the probability that an integer is a prime is
1/logx. Thus, naively, one may expect that the number of primes in an interval
(x,y], for large
x is about
(y−x)/logx, and in a heuristic formula,
π(y)−π(x)∼(y−x)logx=hlogx.(∗)
Here
h=y−x is the length of the interval. This heuristic makes senses only for
h which is much bigger than
logx.
From the Prime Number Theorem (∗) holds if h∼λx, where λ>0 is fixed. From Riemann Hypothesis (∗) holds for h∼x1/2+ϵ for any fixed ϵ>0. (Because the RH gives the error term in the PNT.) There are unconditional results by Huxley and Heath-Brown showing (∗) for h roughly being x7/12.
If h=logxloglogx⋅loglogloglogxlogloglogx, then (∗) fails for a sequence xn→∞. To deal with `small' intervals Selberg worked with almost all x. Namely he considered (∗) for all x∈R+∖S, where |S∩(0,x]|=o(x). In this sense (∗) holds if h/log2x→0 conditionally on RH and for h=x19/77+ϵ unconditionally.
There are also works on the case h∼λlogx. There the distribution of the number of primes on intervals of this size is Poission with parameter λ, conditionally on the Hardy-Littlewood prime tuple conjecture. I think this is due to Gallagher.
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