The average order of an arithmetic function. (English) Zbl 0070.27501

Grosswald, Emil

Duke Math. J. 23, 41-44 (1956).

Let \(F(n)\) denote the number of all prime divisors of \(n\), distinct or not. The author proves that \[ \sum_{_{\substack{ n\leq x\\ 2\nmid n}}} 2^{F(n)}= c_2x\log x-c_5 x+O(x^\varepsilon), \] where \(c_2\) and \(c_5\) are constants and \(c<0.84\). In the proof he uses the property that \(\zeta(s)=O\left(| t|^{1/2(L-1)+\varepsilon}\right)\) holds uniformly for \(\sigma\geq 1-l/2(L-1)\) and any \(\varepsilon>0\), where \(l\) is an integer \(\geq 3\) and \(L=2^{l-1}\). For nonintegral value \(l\) (as the author uses \(l=3.54\ldots\)), the previous result needs a proof.

Reviewer: Loo-Keng Hua (Beijing)

MSC:

11N37

Asymptotic results on arithmetic functions

Keywords:

average order; arithmetic function

Citations:

Zbl 0055.27503