On the distribution of values of an error term related to the Euler function. (English) Zbl 0685.10030

Théorie des nombres, C. R. Conf. Int., Québec/Can. 1987, 785-797 (1989).
[For the entire collection see Zbl 0674.00008.]
Define the error terms H(x) and h(x) by \(\sum_{n\leq x}\phi (n)/n=(6/\pi^ 2)x+H(x)\) and \(h(x)=H(x)-3/\pi^ 2\), where \(\phi\) denotes Euler’s totient function. A well known result of A. Walfisz [Weylsche Exponentialsummen in der neueren Zahlentheorie (Berlin 1963; Zbl 0146.060)] states that \(H(x)\ll (\log x)^{2/3}(\log \log x)^{4/3}\) while more recent work of H. L. Montgomery [Proc. Indian Acad. Sci., Math. Sci. 97, 239-245 (1987; Zbl 0656.10042)] shows that \(H(x)=\Omega_{\pm}(\sqrt{\log \log x}).\) Results of P. Erdős and H. N Shapiro [Can. J. Math. 7, 63-75 (1955; Zbl 0067.276)] imply the existence and continuity of the distribution function \(D(u)=\lim_{x\to \infty}(1/x)\#\{n\leq x |\) h(n)\(\leq u\}.\)
The present author gives further properties of D: (1) D is symmetric, i.e. \(D(u)+D(-u-0)=1\) for all \(u\geq 0\). (2) The moments of all orders of D exist; moreover all odd moments are zero. (3) \(0<D(u)<1\) for all \(u\in {\mathbb{R}}\). As an application of (3) a lower bound in a change-of-sign problem related to H(x) is derived. Finally the same method is applied to the error term associated with \(\sum_{n\leq x}\sigma (n)/n\) where \(\sigma (n)=\sum_{d | n}d\).
Reviewer: J.Herzog


11N37 Asymptotic results on arithmetic functions
11A25 Arithmetic functions; related numbers; inversion formulas