2018年4月6日

# On sums of logarithmic averages of gcd-sum functions

• Isao Kiuchi
• ,

DOI
10.1016/j.jnt.2016.12.021

Let $\gcd(k,j)$ be the greatest common divisor of the integers $k$ and $j$.<br />
For any arithmetical function $f$, we establish several asymptotic formulas for<br />
weighted averages of gcd-sum functions with weight concerning logarithms, that<br />
is $$\sum_{k\leq x}\frac{1}{k} \sum_{j=1}^{k}f(\gcd(k,j)) \log j.$$ More<br />
precisely, we give asymptotic formulas for various multiplicative functions<br />
such as $f=id$, $\phi$, $id_{1+a}$ and $\phi_{1+a}$ with $-1&lt;a&lt;0$. We also<br />
establish some formulas of Dirichlet series having coefficients of the sum<br />
function $\sum_{j=1}^{k}s_{k}(j)\log j$ where $s_{k}(j)$ is Anderson--Apostol<br />
sums.

リンク情報
DOI
https://doi.org/10.1016/j.jnt.2016.12.021
arXiv
http://arxiv.org/abs/arXiv:1804.01902
URL
http://arxiv.org/abs/1804.01902v1
ID情報
• DOI : 10.1016/j.jnt.2016.12.021
• arXiv ID : arXiv:1804.01902

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