A homothetic arithmetic function of ratio $K$ is a function $f:\mathbb{N}\to R$ such that $f\left(Kn\right)=f\left(n\right)$ for every $n\in \mathbb{N}$. Periodic arithmetic funtions are always homothetic, while the converse is not true in general. In this paper we study homothetic and periodic arithmetic functions. In particular we give an upper bound for the number of elements of $f\left(\mathbb{N}\right)$ in terms of the period and the ratio of $f$.

For $1\le c\le p-1$, let $E_1,E_2,\cdots ,E_m$ be fixed numbers of the set $\{0,1\}$, and let $a_1,a_2,\cdots ,a_m$$(1\le a_i\le p...$

The paper deals with lower bounds for the remainder term in asymptotics for a certain class of arithmetic functions. Typically, these are generated by a Dirichlet series of the form ζ 2(s)ζ(2s−1)ζ M(2s)H(s), where M is an arbitrary integer and H(s) has an Euler product which converges absolutely for R s > σ0, with some fixed σ0 < 1/2.

Let $\mathbb{P}=\{p_1,p_2,\cdots ,p_i,\cdots \}$ be the set of prime numbers (or more generally a set of pairwise co-prime elements). Let us denote $A_p^{a,b}=\{p^{an+b}m\mid n\in \mathbb{N}\cup \left\{0\right\};m\in \mathbb{N},p\mathrm{does}\mathrm{not}\mathrm{divide}m\}$, where $a\in \mathbb{N},b\in \mathbb{N}\cup \left\{0\right\}$. Then for arbitrary finite set $B$, $B\subset \mathbb{P}$ holds $$d\left(\bigcap _{p_i\in B}{A}_{p_i}^{a_i,b_i}\right)=\prod _{p_i\in B}d\left(A_{p_i}^{a_i,b_i}\right),$$ and $$d\left(A_{p_i}^{a_i,b_i}\right)=\frac{\frac{1}{p_i^{b_i}}\left(1-\frac{1}{p_i}\right)}{1-\frac{1}{p_i^{a_i}}}.$$ If we denote $$A=\left\{\frac{\frac{1}{p^b}\left(1-\frac{1}{p}\right)}{1-\frac{1}{p^a}}\mid p\in \mathbb{P},a\in \mathbb{N},b\in \mathbb{N}\cup \left\{0\right\}\right\},$$ where $\mathbb{P}$ is the set of all prime numbers, then for closure of set $A$ holds $$\mathrm{cl}A=A\cup B\cup \{0,1\},$$ where $B=\left\{\frac{1}{p^b}\left(1-\frac{1}{p}\right)\mid p\in \mathbb{P},b\in \mathbb{N}\cup \left\{0\right\}\right\}$.

Let $c_q\left(n\right)$ be the Ramanujan sum, i.e. $c_q\left(n\right)={\sum }_{d\left|\right(q,n)}d\mu (q/d)$, where μ is the Möbius function. In a paper of Chan and Kumchev (2012), asymptotic formulas for ${\sum }_{n\le y}{\left({\sum }_{q\le x}{c}_q\left(n\right)\right)}^k$ (k = 1,2) are obtained. As an analogous problem, we evaluate ${\sum }_{n\le y}{\left({\sum }_{n\le x}c{̂}_q\left(n\right)\right)}^k$ (k = 1,2), where $c{̂}_q\left(n\right):={\sum }_{d\left|\right(q,n)}d\left|\mu (q/d)\right|$.