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.

Let $p_1,p_2,p_3$ be three distinct odd primes and let $a\ge 2$, $b\ge 1$ be two integers. All nonnegative integer solutions $\left(x,y,z\right)$ of Pell equation $x^2-\left(a^2{b}^2\pm a\right)y^2=1$ and $y^2-2p_1{p}_2{p}_3{z}^2=4b^2$ were given. Thus, the results of reference [12] were generalized.