On the sum of exponential divisors of an integer. (English) Zbl 0902.11037

Pétermann, Y.-F. S.; Wu, J.

Acta Math. Hung. 77, No. 1-2, 159-175 (1997).

Let \(n\) be a positive integer with canonical decomposition \(n=p_1^{\nu_1}\cdots p_{k}^{\nu_{k}}\). \(d\) is called an exponential divisor of \(n\) if \(d=p_1^{\mu_1}\cdots p_{k}^{\mu_{k}}\) with \(\mu_ j| \nu_ j\) (\(1\leq j\leq k\)). Denote by \(\sigma^{(e)}(n)\) the sum of all positive exponential divisors of \(n>1\). J. Fabrykowski and M. V. Subbarao [Théorie des nombres, C. R. Conf. Int., Québec/Can. 1987, 201-206 (1989; Zbl 0684.10041)] proved that \[ \sum_{n\leq x}\sigma^{(e)}(n)=Dx^ 2+O(x^{1+\epsilon}) \qquad(\forall\epsilon>0), \] where \(D\approx 0.568\) is an effective constant, and \[ \limsup_{n\to \infty}{\sigma^{(e)}(n)\over n \log \log n}={6\over\pi^ 2}e^{\gamma}, \] where \(\gamma\) is the Euler constant. Denote by \(\Delta(x)\) the error term in the asymptotic formula. Then \(\Delta(x)\ll x(\log x)^2\) and \(\Delta(x)=\Omega(x\log\log x)\) can be shown from their results. The authors give a sharper \(O\)-estimate \(\Delta(x)\ll x(\log x)^{5/3}\) and a more precise \(\Omega\)-estimate \(\Delta(x)=\Omega_{\pm}(x\log\log x)\). Their proof is based on the estimate \(\Delta(a, b; x)\) in the asymptotic formula \[ \sum_{mn^ a\leq x}mn^ a={1\over 2}\zeta(2a-b)x^2+\Delta(a, b; x), \] where \(a\), \(b\) are positive real numbers with \(b\leq a-1\) and \(\zeta\) is the Riemann zeta-function.

Reviewer: Takao Komatsu (Nagaoka, Niigata)

MSC:

11N37

Asymptotic results on arithmetic functions

Keywords:

exponential divisor; \(O\)-estimate; \(\Omega\)-estimate; arithmetic functions; asymptotic formula

Citations:

Zbl 0684.10041