Weighted Average Number of Prime $m$-tuples lying on an Admissible $k$-tuple of Linear Forms

Abstract

We find an upper bound for the sum ${\sum }_{x<n\le 2x}{\text{1}}_{\mathbb{P}}(n+h_{i_1})\cdots {\text{1}}_{\mathbb{P}}(n+h_{i_{m+1}})w_n$, where $(h_{i_1},...,h_{i_{m+1}})$ is any $(m+1)$-tuple of elements in the admissible set $\mathcal{H}=\{h_1,...,h_k\}$, $m\ge 1$ and $x$ is sufficiently large, with the same weights $w_n$ used in the Maynard's paper "Dense clusters of primes in subsets". The estimate will be uniform over positive integer $k$ with $m+1\le k\le (\log x{)}^{1/5}$ and on admissible set $\mathcal{H}$ with $0\le h_1<...<h_k\le x$. The upper bound will depend on an integral of a smooth function and on the singular series of $\mathcal{H}$, which naturally arises in this context.

arXiv comment14 pages. Any kind of comment is welcome