@article{22192,
  abstract     = {In 1976, Gallagher showed that the Hardy–Littlewood conjectures on prime k-tuples imply that the
distribution of primes in log-size intervals is Poissonian. He did so by computing average values
of the singular series constants over different sets of a fixed size k contained in an interval [1,h]
as h → ∞, and then using this average to compute moments of the distribution of primes. In this
paper, we study averages where k is relatively large with respect to h. We then apply these averages
to the tail of the distribution. For example, we show, assuming appropriate Hardy–Littlewood
conjectures and in certain ranges of the parameters, the number of intervals [n,n + λlogx] with
n ≤ x containing at least k primes is ≪ x exp(−k/(λe)).},
  author       = {Kuperberg, Vivian Zieve},
  issn         = {1464-3847},
  journal      = {The Quarterly Journal of Mathematics},
  number       = {4},
  pages        = {1457--1479},
  publisher    = {Oxford University Press},
  title        = {{Sums of singular series with large sets and the tail of the distribution of primes}},
  doi          = {10.1093/qmath/haad030},
  volume       = {74},
  year         = {2023},
}