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Let Λ denote the von Mangoldt function, and (n, q) be the greatest common divisor of positive integers n and q. For any positive real numbers x and y, we shall consider several asymptotic formulas for sums of sums involving the von Mangoldt function; S k (x , y) : = ∑ n ≤ y ∑ q ≤ x ∑ d | (n , q) d Λ q d k for k = 1 , 2 . [ABSTRACT FROM AUTHOR]

Let (n, q) denote the greatest common divisor of positive integers n and q, and let f r denote the characteristic function of r-full numbers. We consider several asymptotic formulas for sums of the modified square-full ( r = 2 ) and cube-full numbers ( r = 3 ), which is ∑ n ≤ y ∑ q ≤ x ∑ d | (n , q) d f r q d log x q with any positive real numbers x and y. Moreover, we derive the asymptotic formula of the above with r = 2 under the Riemann Hypothesis. [ABSTRACT FROM AUTHOR]

Let f 3 denote the characteristic function of cube-full numbers, and let (n, q) be the greatest common divisor of positive integers n and q. For any positive real numbers x and y, we shall consider several asymptotic formulas for sums of sums of modified cube-full numbers, which is ∑ n ≤ y ∑ q ≤ x ∑ d | (n , q) d f 3 q / d k with k = 1 , 2 . [ABSTRACT FROM AUTHOR]

Let gcd (k , j) denote the greatest common divisor of the integers k and j, and let r be any fixed positive integer. Define M r (x ; f) : = ∑ k ≤ x 1 k r + 1 ∑ j = 1 k j r f (gcd (j , k)) for any large real number x ≥ 5 , where f is any arithmetical function. Let ϕ , and ψ denote the Euler totient and the Dedekind function, respectively. In this paper, we refine asymptotic expansions of M r (x ; id) , M r (x ; ϕ) and M r (x ; ψ) . Furthermore, under the Riemann Hypothesis and the simplicity of zeros of the Riemann zeta-function, we establish the asymptotic formula of M r (x ; id) for any large positive number x > 5 satisfying x = [ x ] + 1 2 . [ABSTRACT FROM AUTHOR]

Let gcd (k , j) be the greatest common divisor of the integers k and j. For any arithmetic function f, we establish several asymptotic formulas for weighted averages of gcd-sum functions with weight concerning logarithms, that is ∑ k ≤ x 1 k ∑ j = 1 k f (gcd (k , j)) log j. More precisely, we give asymptotic formulas for various multiplicative functions such as f = id , ϕ , id 1 + a and ϕ 1 + a with - 1 < a < 0 . We also consider several formulas of Dirichlet series associated with Anderson–Apostol sums. [ABSTRACT FROM AUTHOR]

For $-1\leq a \leq 0$ let ${\it\Delta} _{a}(x;r)$ denote the error term in the asymptotic formula for $\sum \limits _{n \leq x} {\it\sigma} _{a}(n)e(rn)$ . We study the mean square of the function $|{\it\Delta} _{a}(x+U;r)-{\it\Delta} _{a}(x;r)|$ for short intervals. [ABSTRACT FROM AUTHOR]