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Let $\gcd(d_{1},\ldots,d_{k})$ be the greatest common divisor of the positive integers $d_{1},\ldots,d_{k}$, for any integer $k\geq 2$, and let $\tau$ and $\mu$ denote the divisor function and the M\"{o}bius function, respectively. For an arbitrary arithmetic function $g$ and for any real number $x>5$ and any integer $k\geq 3$, we define the sum $$ S_{g,k}(x) :=\sum_{n\leq x}\sum_{d_{1}\cdots d_{k}=n} g(\gcd(d_{1},\ldots,d_{k})) $$ In this paper, we give asymptotic formulas for $S_{\tau,k}(x)$ and $S_{\mu,k}(x)$ for $k\geq 3$., Comment: 13 pages

Let gcd(k,j) denote the greatest common divisor of the integers k and j, and let r be any fixed positive integer. Define Mr(x;f):=kx1kr+1j=1kjrf(gcd(j,k)) for any large real number x5, where f is any arithmetical function. Let , and denote the Euler totient and the Dedekind function, respectively. In this paper, we refine asymptotic expansions of Mr(x;id), Mr(x;) and Mr(x;). Furthermore, under the Riemann Hypothesis and the simplicity of zeros of the Riemann zeta-function, we establish the asymptotic formula of Mr(x;id) for any large positive number x>5 satisfying x=[x]+12. (VLID)6034101 Version of record

Let ψ denote the Dedekind totient function by id ∗ μ^2, where μ is the Möbius function. We establish the asymptotic formula of the general kth Riesz mean of the arithmetical function n/ψ(n) for any positive integer k ≥ 2; namely 1/k!Σ__〈n ≤x〉n/(ψ(n))(1-n/x)^k=(ζ(4)h(1))/( (k + 1)!ζ(2))x+O(1/kexp(-C((logx)^〈1/3〉)/((log logx)^〈1/3〉))) with h(1)=П__p(1+1/(p^3(1+1/p))-1/(p^4(1+1/p))). Assuming that the Riemann Hypothesis is true, then the error term of the above function can be estimated as O(k^〈−1〉x^〈−1/(2+δ)〉 for any small positive constant δ.

Let $\mathbb{Z}_{m}$ 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 ${\mathbb{Z}}_{m}\times {\mathbb{Z}}_{n}$, respectively. Define $$ \widetilde{D}_{s}(x) = \sum_{mn\leq x}s(m,n)\log\frac{x}{mn} \quad \quad \widetilde{D}_{c}(x) = \sum_{mn\leq x}c(m,n)\log\frac{x}{mn}. $$ In this paper, we study the asymptotic behaviour of functions $\widetilde{D}_{s}(x)$ and $\widetilde{D}_{c}(x)$., Comment: 9 pages

Let $\gcd(k,j)$ be the greatest common divisor of the integers $k$ and $j$. For any arithmetical function $f$, we establish several asymptotic formulas for weighted averages of gcd-sum functions with weight concerning logarithms, that is $$\sum_{k\leq x}\frac{1}{k} \sum_{j=1}^{k}f(\gcd(k,j)) \log j.$$ More precisely, we give asymptotic formulas for various multiplicative functions such as $f=id$, $\phi$, $id_{1+a}$ and $\phi_{1+a}$ with $-1, Comment: 21 Pages

Let $\gcd(j,k)$ be the greatest common divisor of the integers $j$ and $k$. In this paper, we give several interesting asymptotic formulas for weighted averages of the $\gcd$-sum function $f(\gcd(j,k)) $ and the function $\sum_{d|k, d^{s}|j}(f*��)(d) $ for any positive integers $j$ and $k$, namely $$ \sum_{k\leq x}\frac{1}{k^{r+1}}\sum_{j=1}^{k}j^{r}f(\gcd(j,k)) \quad \text{and} \quad \sum_{k\leq x}\frac{1}{k^{s(r+1)}}\sum_{j=1}^{k^s}j^{r} \sum_{\substack{d|k d^{s}|j}}(f*��)(d), $$ with any fixed integer $s> 1$ and any arithmetical function $f$. We also establish mean value formulas for the error terms of asymptotic formulas for partial sums of $\gcd$-sum functions $f(\gcd(j,k)). $, 30 pages