## Tsukuba Journal of Mathematics

- Tsukuba J. Math.
- Volume 12, Number 1 (1988), 241-248.

### Existence of all the asymptotic $\lambda$-th means for certain arithmetical convolutions

#### Abstract

Let $E$ designate either of the classical error terms for the summatory functions of the arithmetical functions $\phi(n)/n$ and $\sigma(n)/n$ ($\phi$ is Euler's function and $\sigma$ the divisor function). By following an idea of Codecà's [3] and by refining some of his estimates we prove that $|E|$ has asymptotic $\lambda$-th order means for all positive real numbers $\lambda$. We also prove that $E$ has asymptotic $k$-th order means for all positive integers $k$, and that this mean is zero whenever $k$ is odd. The results obtained can be applied to functions other than $E$ as well, such as the functions $P$ and $Q$ of Hardy and Littlewood[8], or the divisor functions $G_{-1,k}$[9].

#### Article information

**Source**

Tsukuba J. Math., Volume 12, Number 1 (1988), 241-248.

**Dates**

First available in Project Euclid: 30 May 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.tkbjm/1496160645

**Digital Object Identifier**

doi:10.21099/tkbjm/1496160645

**Mathematical Reviews number (MathSciNet)**

MR949911

**Zentralblatt MATH identifier**

0661.10056

#### Citation

Y-F.S, Petermann. Existence of all the asymptotic $\lambda$-th means for certain arithmetical convolutions. Tsukuba J. Math. 12 (1988), no. 1, 241--248. doi:10.21099/tkbjm/1496160645. https://projecteuclid.org/euclid.tkbjm/1496160645