On Sums of Sums Involving the Von Mangoldt Function

Let \(\Lambda \) $\mathrm{\Lambda }$ denote the von Mangoldt function, and (n, q) be the greatest common divisor of positive integers n and q. For any positive real numbers x and y, we shall consider several asymptotic formulas for sums of sums involving the von Mangoldt function; \( S_{k}(x,y):=\sum _{n\le y}\left( \sum _{q\le x}\right. \left. \sum _{d|(n,q)}d\Lambda \left( \frac{q}{d}\right) \right) ^{k} \) $S_k(x,y):={\sum }_{n\le y}\left({\sum }_{q\le x}\right){\left({\sum }_{d|(n,q)},d,\mathrm{\Lambda },\left(\frac{q}{d}\right)\right)}^k$ for \(k=1,2\) $k=1,2$ .