论文标题
GCD-SUM函数的平均值II
Sums of averages of gcd-sum functions II
论文作者
论文摘要
令$ \ gcd（k，j）$表示整数$ k $和$ j $的最大共同除数，然后让$ r $成为任何固定的正整数。定义$$ m_r（x; f）：= \ sum_ {k \ leq x} \ frac {1} {k^{r+1}} \ sum_ {j = 1}^{k}^{k} j^{j^{r}令$ ϕ $和$ψ$分别表示Euler的构件和Dedekind函数。在本文中，我们完善了$ m_r（x; {\ rm ID}）$，$ m_r（x; ϕ）$和$ m_r（x;ψ）$的渐近扩展。此外，在Riemann假设和Riemann Zeta功能的零的简单下，我们建立了$ M_R（x; {\ rm ID}）$的渐近公式，用于任何大型正数$ x> 5 $ x> 5 $满足$ x = [x]+x = [x]+\ frac {x]+\ frac {frac {1} $ {2} $。
Let $\gcd(k,j)$ denote the greatest common divisor of the integers $k$ and $j$, and let $r$ be any fixed positive integer. Define $$ M_r(x; f) := \sum_{k\leq x}\frac{1}{k^{r+1}}\sum_{j=1}^{k}j^{r}f(\gcd(j,k)) $$ for any large real number $x\geq 5$, 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 $M_r(x; {\rm id})$, $M_r(x;ϕ)$ 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 $M_r(x;{\rm id})$ for any large positive number $x>5$ satisfying $x=[x]+\frac{1}{2}$.
