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

TL;DR

This paper investigates the behavior of prime distributions in large sets by analyzing averages of singular series constants, extending Gallagher's work, and applying results to estimate the tail probabilities of prime counts in short intervals.

Contribution

It extends Gallagher's analysis to larger sets and applies these averages to estimate the tail of the prime distribution in short intervals under Hardy--Littlewood conjectures.

Findings

Average singular series for large sets studied

Derived bounds on intervals with many primes

Estimated tail probabilities of prime counts

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 $h \to \infty$, 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 $\ll x\exp(-k/(\lambda e)).$