Asymptotic formulas for Dirichlet convolutions

In this paper, we use convoluting prime-number-theorem-related functions (Mobius, von Mangoldt, and Liouville) by the Piltz divisor function to research an asymptotic formula for the convolution sum $ (x\ge 1) $. Our main result is Theorem 1, which separates the error terms for the PNT-related functions and the Piltz divisor function. Under the assumption of the conjectural estimate for the latter, we obtain the expected error term as in PNT.