PRIMES IN SHORT INTERVALS

arXiv:math/0409258v1 [math.NT] 15 Sep 2004

PRIMES IN SHORT INTERVALS

Hugh L. Montgomery1 K. Soundararajan2 Dedicated to Freeman Dyson, with best wishes on the occasion of his eightieth birthday. Abstract. Contrary to what would be predicted on the basis of Cram´ er’s model concerning the distribution of prime numbers, we develop evidence that the distribution of ψ(x + H) − ψ(x), for 0 ≤ x ≤ N, is approximately normal with mean ∼ H and variance ∼ H log N/H, when N δ ≤ H ≤ N 1−δ .

0. Introduction Cram´er [4] modeled the distribution of prime numbers by independent random variables Xn (for n ≥ 3) that take the value 1 (n is “prime”) with probability 1/ log n and take the value 0 (n is “composite”) with probability 1 − 1/ log n. If pn denotes the nth prime number this model predicts that 1 card{n : 1 ≤ n ≤ N, pn+1 − pn > c log pn } = e−c N→∞ N lim

for all fixed positive real numbers c. Gallagher [6] showed that the above follows from Hardy & Littlewood’s [10, p. 61] quantitative version of the prime k-tuple conjecture: If D = {d1 , d2 , . . . , dk } is a set of k distinct integers, then (1)

k X Y

Λ(n + di ) = (S(D) + o(1))x

n≤x i=1

as x → ∞ where S(D) is the singular series (2)

S(D) =

X

q1 ,... ,qk 1≤qi 1. Then

X

a1 ,... ,ak 0