MISC

2018年1月11日

Sums of weighted averages of gcd-sum functions II

  • Isao Kiuchi
  • ,
  • Sumaia Saad Eddin

記述言語
掲載種別
機関テクニカルレポート,技術報告書,プレプリント等

In this paper, we establish the following two identities involving the Gamma<br />
function and Bernoulli polynomials, namely $$ \sum_{k\leq x}\frac{1}{k^s}<br />
\sum_{j=1}^{k^s}\log\Gamma\left(\frac{j}{k^s}\right)<br />
\sum_{\substack{d|k \\ d^{s}|j } }f*\mu(d) \quad {\rm and } \quad \sum_{k\leq<br />
x}\frac{1}{k^s}\sum_{j=0}^{k^{s}-1}<br />
B_{m}\sum_{\substack{d|k \\ d^{s}|j } } f*\mu(d) $$ with any fixed integer $s&gt;<br />
1$ and any arithmetical function $f$. We give asymptotic formulas for them with<br />
various multiplicative functions $f$. We also consider several formulas of<br />
Dirichlet series associated with the above identities. This paper is a<br />
continuation of an earlier work of the authors.

リンク情報
arXiv
http://arxiv.org/abs/arXiv:1801.03653
URL
http://arxiv.org/abs/1801.03653v2
ID情報
  • arXiv ID : arXiv:1801.03653

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