The Riemann hypothesis via the generalized von Mangoldt function

Based on work previously done by Gonek, Graham, and Lee, we show that the Riemann Hypothesis (RH) can be reformulated in terms of certain asymptotic estimates for twisted sums with k-fold convolution of von Mangoldt function and the generalized von Mangoldt function. For each $k \in\mathbb{N}$, we study two types of twisted sums:

1. $\sum_{n\leq x} \Lambda^k(n)n^{-iy}$, where $\Lambda^k(n) = \underbrace{\Lambda\star\cdots\Lambda}_\text{k copies}$
2. $\sum_{n\leq x} \Lambda_k(n)n^{-iy}$, where $\Lambda_k(n) :=\sum_{d|n}\mu(d)\left(\log{\frac{n}{d}}\right)^k$.

Where $\Lambda$ is the von Mangoldt function and $\mu$ is the Möbius function, and establish similar connections with RH.