We assign to each pair of positive integers $n$ and $k\ge 2$ a digraph $G(n,k)$ whose set of vertices is $H=\{0,1,\cdots ,n-1\}$ and for which there is a directed edge from $a\in H$ to $b\in H$ if $a^k\equiv b(modn)$. We investigate the structure of $G(n,k)$. In particular, upper bounds are given for the longest cycle in $G(n,k)$. We find subdigraphs of $G(n,k)$, called fundamental constituents of $G(n,k)$, for which all trees attached to cycle vertices are isomorphic.

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 $f$...

Let $p$ be an odd prime, and let $a$ be an integer not divisible by $p$. When $m$ is a positive integer with $p\equiv 1(mod2m)$ and $2$ is an $m$th power residue modulo $p$, we determine the value of the product ${\prod }_{k\in R_m\left(p\right)}(1+tan(\pi ak/p))$, where $$R_m\left(p\right)=\{0<k<p:k\in \mathbb{Z}\text{is}\text{an}m\text{th}\text{power}\text{residue}\text{modulo}p\}.$$ In particular, if $p=x^2+64y^2$ with $x,y\in \mathbb{Z}$, then $$\prod _{k\in R_4\left(p\right)}\left(1+tan\pi \frac{ak}{p}\right)={(-1)}^y{(-2)}^{(p-1)/8}.$$

In the first part of the paper we prove that the Zeckendorf sum-of-digits function $s_z\left(n\right)$ and similarly defined functions evaluated on polynomial sequences of positive integers or primes satisfy a central limit theorem. We also prove that the Zeckendorf expansion and the $q$-ary expansions of integers are asymptotically independent.

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.

In the 1990s, J.C. Tong gave a sharp upper bound on the minimum of $k$ consecutive approximation constants for the nearest integer continued fractions. We generalize this to the case of approximation by Rosen continued fraction expansions. The Rosen fractions are an infinite set of continued fraction algorithms, each giving expansions of real numbers in terms of certain algebraic integers. For each, we give a best possible upper bound for the minimum in appropriate consecutive blocks of approximation...