## THE DISTRIBUTION OF PRIME NUMBERS (2006)

Citations: | 1 - 0 self |

### BibTeX

@MISC{Soundararajan06thedistribution,

author = {K. Soundararajan},

title = {THE DISTRIBUTION OF PRIME NUMBERS},

year = {2006}

}

### OpenURL

### Abstract

What follows is an expanded version of my lectures at the NATO School on Equidistribution. I have tried to keep the informal style of the lectures. In particular, I have sometimes oversimplified matters in order to convey the spirit of an argument. Lecture 1: The Cramér model and gaps between consecutive primes The prime number theorem tells us that π(x), the number of primes below x, is ∼ x / logx. Equivalently, if pn denotes the n-th smallest prime number then pn ∼ n log n. What is the distribution of the gaps between consecutive primes, pn+1 − pn? We have just seen that pn+1 − pn is approximately log n “on average”. How often do we get a gap of size 2 logn, say; or of size 1 log n? One way to make this question precise 2 is to fix an interval [α, β] (with 0 ≤ α < β) and ask for