Abstract
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 \(n {\leqslant } x\) for which, for each \(j {\leqslant } r\), the interval \((n, n + (\lambda _1 + \cdots + \lambda _j)\log x]\) contains exactly m 1 + ⋯ + m j primes, is asymptotically equal to \(\prod _{j = 1}^r( {\mathrm {e}}^{-\lambda _j}\lambda ^{m_j}/m_j!)\) 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.
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The author is grateful to the anonymous referee, for carefully reading this manuscript and making corrections.
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Freiberg, T. (2018). A Note on the Distribution of Primes in Intervals. In: Pintz, J., Rassias, M. (eds) Irregularities in the Distribution of Prime Numbers. Springer, Cham. https://doi.org/10.1007/978-3-319-92777-0_2
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DOI: https://doi.org/10.1007/978-3-319-92777-0_2
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