## 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

### MSC:

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

### Citations:

Zbl 0674.00008; Zbl 0146.060; Zbl 0656.10042; Zbl 0067.276