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

The article investigates the behavior of averages of singular series constants associated with prime k-tuples, particularly when the size k of the tuples grows with respect to the interval parameter h. Building on Gallagher's 1976 work linking the Hardy–Littlewood prime k-tuple conjectures to the Poisson distribution of primes in logarithmic intervals, the paper extends these results to larger k, providing bounds on sums of singular series and moments of prime counts in intervals of size proportional to \(\log x\). Assuming a uniform version of the Hardy–Littlewood conjectures, it establishes asymptotic formulas for moments of the prime counting function and derives exponential upper bounds on the frequency of intervals containing at least k primes, where k grows slowly with x. Additionally, unconditional bounds are obtained via the Selberg sieve, yielding weaker but still nontrivial estimates on the tail distribution of primes in short intervals. The work also formulates conjectures on the validity of Poissonian behavior for larger k and discusses the gap between known upper and lower bounds in this regime.