On sums of arithmetic functions involving the greatest common divisor

Abstract

Let $\gcd (d_1,\dots ,d_k)$ be the greatest common divisor of the positive integers $d_1,\dots ,d_k$, for any integer $k\ge 2$, and let $\tau$ and $\mu$ denote the divisor function and the Möbius function, respectively. For an arbitrary arithmetic function $g$ and for any real number $x>5$ and any integer $k\ge 3$, we define the sum $$S_{g,k}(x):=\sum _{n\le x}\sum _{d_1\cdots d_k=n}g(\gcd (d_1,\dots ,d_k))$$ In this paper, we give asymptotic formulas for $S_{\tau ,k}(x)$ and $S_{\mu ,k}(x)$ for $k\ge 3$.

13 pages