当前位置: X-MOL 学术Results Math. › 论文详情
Our official English website, www.x-mol.net, welcomes your feedback! (Note: you will need to create a separate account there.)
Sums of Averages of GCD-Sum Functions II
Results in Mathematics ( IF 2.2 ) Pub Date : 2021-02-14 , DOI: 10.1007/s00025-021-01357-x
Lisa Kaltenböck , Isao Kiuchi , Sumaia Saad Eddin , Masaaki Ueda

Let \( \gcd (k,j) \) denote the greatest common divisor of the integers k and j, and let r be any fixed positive integer. Define

$$\begin{aligned} M_r(x; f) := \sum _{k\le x}\frac{1}{k^{r+1}}\sum _{j=1}^{k}j^{r}f(\gcd (j,k)) \end{aligned}$$

for any large real number \(x\ge 5\), where f is any arithmetical function. Let \(\phi \), and \(\psi \) denote the Euler totient and the Dedekind function, respectively. In this paper, we refine asymptotic expansions of \(M_r(x; \mathrm{id})\), \(M_r(x;{\phi })\) and \(M_r(x;{\psi })\). Furthermore, under the Riemann Hypothesis and the simplicity of zeros of the Riemann zeta-function, we establish the asymptotic formula of \(M_r(x;\mathrm{id})\) for any large positive number \(x>5\) satisfying \(x=[x]+\frac{1}{2}\).



中文翻译:

GCD-Sum函数平均值的和II

\(\ gcd(k,j)\)表示整数kj的最大公约数,令r为任何固定的正整数。定义

$$ \ begin {aligned} M_r(x; f):= \ sum _ {k \ le x} \ frac {1} {k ^ {r + 1}} \ sum _ {j = 1} ^ {k} j ^ {r} f(\ gcd(j,k))\ end {aligned} $$

对于任何大的实数\(x \ ge 5 \),其中f是任何算术函数。令\(\ phi \)\(\ psi \)分别表示Euler totient和Dedekind函数。在本文中,我们细化了\(M_r(x; \ mathrm {id})\)\(M_r(x; {\ phi})\)\(M_r(x; {\ psi})\ )。此外,根据黎曼假设和黎曼zeta函数零的简单性,我们为任何大正数\(x> 5 \)建立\(M_r(x; \ mathrm {id})\)的渐近公式满足\(x = [x] + \ frac {1} {2} \)

更新日期:2021-02-15
down
wechat
bug