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 $f$...

The aim of this work is to estimate exponential sums of the form ${\sum }_{n\le x}\Lambda \left(n\right)exp\left(2i\pi (f\left(n\right)+\beta n)\right)$, where Λ denotes von Mangoldt’s function, f a digital function, and β ∈ ℝ a parameter. This result can be interpreted as a Prime Number Theorem for rotations (i.e. a Vinogradov type theorem) twisted by digital functions.

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.

We investigate properties of coset topologies on commutative domains with an identity, in particular, the 𝓢-coprime topologies defined by Marko and Porubský (2012) and akin to the topology defined by Furstenberg (1955) in his proof of the infinitude of rational primes. We extend results about the infinitude of prime or maximal ideals related to the Dirichlet theorem on the infinitude of primes from Knopfmacher and Porubský (1997), and correct some results from that paper. Then we determine cluster...