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Let $\gcd (n_{1},\ldots ,n_{k})$ denote the greatest common divisor of positive integers $n_{1},\ldots ,n_{k}$ and let $\phi $ be the Euler totient function. For any real number $x>3$ and any integer $k\geq 2$ , we investigate the asymptotic behaviour of $\sum _{n_{1}\ldots n_{k}\leq x}\phi (\gcd (n_{1},\ldots ,n_{k})). $ [ABSTRACT FROM AUTHOR]

Let gcd (j , k) denote the greatest common divisor of integers k and j. Define A f (x) = ∑ k ≤ x 1 k ∑ j = 1 k f (gcd (j , k)) , for any arithmetical function f. We shall derive several asymptotic expansions of the error terms E f (x) for the partial sums of A f (k) with f = id , id 1 + a (− 1 < a < 0) , ϕ , and ψ , where ϕ and ψ are the Euler totient function and the Dedekind function, respectively. Furthermore, we establish several formulas concerning the sum function A f (x) , and deduce several related results. [ABSTRACT FROM AUTHOR]

Let ℤ m be the additive group of residue classes modulo m. For any positive integers m and n , let s (m , n) and c (m , n) denote the total number of subgroups and cyclic subgroups of the group ℤ m × ℤ n , respectively. Define D ̃ s (x) = ∑ m n ≤ x s (m , n) log x m n and D ̃ c (x) = ∑ m n ≤ x c (m , n) log x m n. In this paper, we study the asymptotic behavior of the functions D ̃ s (x) and D ̃ c (x). [ABSTRACT FROM AUTHOR]