Sums of singular series with large sets and the tail of the distribution of primes

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 $h\to \mathrm{\infty }$, 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 $[n,n+\lambda \log x]$ with $n\le x$ containing at least $k$ primes is $\ll x\exp (-k\mathrm{/}(\lambda e))\mathrm{.}$