A sequence is called $k$-automatic if the $n$’th term in the sequence can be generated by a finite state machine, reading $n$ in base $k$ as input. We show that for many multiplicative functions, the sequence ${\left(f\left(n\right)\text{mod}v\right)}_{n\ge 1}$ is not $k$-automatic. Among these multiplicative functions are ${\gamma }_m\left(n\right),{\sigma }_m\left(n\right),\mu \left(n\right)$ et $\phi \left(n\right)$.

Let $n$ be a positive integer, and $C_n\left(r\right)$ the set of all $n\times n$ $r$-circulant matrices over the Boolean algebra $B=\{0,1\}$, $G_n={\bigcup }_{r=0}^{n-1}{C}_n\left(r\right)$. For any fixed $r$-circulant matrix $C$ ($C\ne 0$) in $G_n$, we define an operation “$*$” in $G_n$ as follows: $A*B=ACB$ for any $A,B$ in $G_n$, where $ACB$ is the usual product of Boolean matrices. Then $(G_n,*)$ is a semigroup. We denote this semigroup by $G_n\left(C\right)$ and call it the sandwich semigroup of generalized circulant Boolean matrices with sandwich matrix $C$. Let $F$ be an idempotent element in $G_n\left(C\right)$ and $M\left(F\right)$ the maximal subgroup...

Let $p$ be an odd prime and $c$ a fixed integer with $(c,p)=1$. For each integer $a$ with $1\le a\le p-1$, it is clear that there exists one and only one $b$ with $0\le b\le p-1$ such that $ab\equiv c$ (mod $p$). Let $N(c,p)$ denote the number of all solutions of the congruence equation $ab\equiv c$ (mod $p$) for $1\le a$, $b\le p-1$ in which $a$ and $\overline{b}$ are of opposite parity, where $\overline{b}$ is defined by the congruence equation $b\overline{b}\equiv 1(modp)$. The main purpose of this paper is to use the properties of Dedekind sums and the mean value theorem for Dirichlet $L$-functions to study the hybrid mean value...