We establish a general mean value result for arithmetic functions over short intervals with the Selberg-Delange method. As an application, we generalize the Deshouillers-Dress-Tenenbaum arcsine law on divisors to the short interval case.

The present paper deals with the summatory function of functions acting on the digits of an $q$-ary expansion. In particular let $n$ be a positive integer, then we call$$\begin{array}{c}n=\sum _{r=0}^{\ell }{d}_r\left(n\right)q^r\text{with}{d}_r\left(n\right)\in \{0,...,q-1\}\end{array}$$its $q$-ary expansion. We call a function $f$strictly $q$-additive, if for a given value, it acts only on the digits of its representation, i.e.,$$f\left(n\right)=\sum _{r=0}^{\ell }f\left(d_r\left(n\right)\right).$$Let $p\left(x\right)={\alpha }_0{x}^{{\beta }_0}+\cdots +{\alpha }_d{x}^{{\beta }_d}$ with ${\alpha }_0,{\alpha }_1,...,{\alpha }_d,\in ℝ$, ${\alpha }_0\>0$, ${\beta }_0\>\cdots \>{\beta }_d\ge 1$ and at least one ${\beta }_i\notin \mathbb{Z}$. Then we call $p$ a pseudo-polynomial.The goal is to prove that for a $q$-additive function $f$ there exists an $\epsilon \>0$ such that$$\begin{array}{c}\sum _{n\le N}f\left(⌊p\left(n\right)⌋\right)={\mu }_{f}N{log}_q\left(p\left(N\right)\right)\hfill \\ \hfill +N{F}_{f,{\beta }_0}\left({log}_q\left(p\left(N\right)\right)\right)+𝒪\left(N^{1-\epsilon }\right),\end{array}$$where ${\mu }_f$ is the mean of the values of $f$...

1. Summary. In a sequence of three papers we study the circle problem and its generalization involving the logarithmic mean. Most of the deeper results in this area depend on estimates of exponential sums. For the circle problem itself Chen has carried out such estimates using three two-dimensional Weyl steps with complicated techniques. We make the same Weyl steps but our approach is simpler and clearer. Crucial is a good understanding of the Hessian determinant that appears and a simple...

1. Summary. In Part II we study arithmetic functions whose Dirichlet series satisfy a rather general type of functional equation. For the logarithmic Riesz mean of these functions we give a representation involving finite trigonometric sums. An essential tool here is the saddle point method. Estimation of the exponential sums in the special case of the circle problem will be the topic of Part III.