A Note on the Distribution of Primes in Intervals

Assuming a certain form of the Hardy–Littlewood prime tuples conjecture, we show that, given any positive numbers λ 1, …, λ r and nonnegative integers m 1, …, m r, the proportion of positive integers n ≤ x $$n {\leqslant } x$$ for which, for each j ≤ r $$j {\leqslant } r$$ , the interval ( n , n + ( λ 1 + ⋯ + λ j ) log x ] $$(n, n + (\lambda _1 + \cdots + \lambda _j)\log x]$$ contains exactly m 1 + ⋯ + m j primes, is asymptotically equal to ∏ j = 1 r ( e − λ j λ m j ∕ m j ! ) $$\prod _{j = 1}^r( {\mathrm {e}}^{-\lambda _j}\lambda ^{m_j}/m_j!)$$ as x →∞. This extends a result of Gallagher, who considered the case r = 1. We use a direct inclusion–exclusion argument in place of Gallagher’s moment calculation, thereby avoiding recourse to the moment determinacy property of Poisson distributions.