The Distribution of Prime Numbers
نویسندگان
چکیده
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 logn. What is the distribution of the gaps between consecutive primes, pn+1 − pn? We have just seen that pn+1 − pn is approximately logn “on average”. How often do we get a gap of size 2 logn, say; or of size 1 2 logn? One way to make this question precise is to fix an interval [α, β] (with 0 ≤ α < β) and ask for
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