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 $\sum_{x<n\leq 2x}\textbf{1}{\mathbb{P}}(n+h{i_{1}})\cdots\textbf{1}{\mathbb{P}}(n+h{i_{m+1}})w_{n}$, where $(h_{i_{1}},...,h_{i_{m+1}})$ is any $(m+1)$-tuple of elements in the admissible set $\mathcal{H}={h_{1},...,h_{k}}$, $m\geq 1$ 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 $m+1\leq k\leq (\log x)^{1/5}$ and on admissible set $\mathcal{H}$ with $0\leq h_{1}<...<h_{k}\leq x$. 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.