The sifted von Mangoldt weight is nonnegative

The sifted von Mangoldt weight $A_N(n) = \Lambda(n)\,\mathbf 1_{(n,\,\sqrt N\sharp)=1}$, which removes from $\Lambda$ every $n$ sharing a prime factor with the primorial of $\sqrt N$, is nonnegative for all $N$ and $n$. It is either $\Lambda(n)$, which is nonnegative, or zero.

This is the weight appearing in equation (8.10) and, with the modulus $q_0 = \sqrt N \sharp$, in the exponential sums $S_{\eta,q_0}$ that Theorem 1.3 estimates.