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 $a_1,a_2,\cdots ,a_m$ $(1\le a_i\le p$, $i=1,2,\cdots ,m)$ be of opposite parity with $E_1,E_2,\cdots ,E_m$ respectively such that $a_1{a}_2\cdots a_m\equiv c(modp)$. Let $$N(c,m,p)=\frac{1}{2^{m-1}}\underset{a_1{a}_2\cdots a_m\equiv c(modp)}{\sum _{a_1=1}^{p-1}\sum _{a_2=1}^{p-1}\cdots \sum _{a_m=1}^{p-1}}(1-{(-1)}^{a_1+E_1})(1-{(-1)}^{a_2+E_2})\cdots (1-{(-1)}^{a_m+E_m}).$$ We are interested in the mean value of the sums $$\sum _{c=1}^{p-1}{E}^2(c,m,p),$$ where $E(c,m,p)=N(c,m,p)-\left({(p-1)}^{m-1}\right)/\left(2^{m-1}\right)$ for the odd prime $p$ and any integers $m\ge 2$. When $m=2$, $c=1$, it is the Lehmer problem. In this paper, we generalize the Lehmer problem and use analytic method to give an interesting asymptotic formula of the generalized Lehmer problem.

$G(3,m,n)$ is the group presented by $\langle a,b\mid a^5={\left(ab\right)}^2=b^{m+3}{a}^{-n}{b}^m{a}^{-n}=1\rangle$. In this paper, we study the structure of $G(3,m,n)$. We also give a new efficient presentation for the Projective Special Linear group $PSL(2,5)$ and in particular we prove that $PSL(2,5)$ is isomorphic to $G(3,m,n)$ under certain conditions.

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 $k$ be a finite extension of ${\mathbb{Q}}_p$, let $k_1$, respectively $k_3$, be the division fields of level $1$, respectively $3$, arising from a Lubin-Tate formal group over $k$, and let $\Gamma =$ Gal($k_3/k_1$). It is known that the valuation ring $k_3$ cannot be free over its associated order $𝔄$ in $K\Gamma$ unless $k={\mathbb{Q}}_p$. We determine explicitly under the hypothesis that the absolute ramification index of $k$ is sufficiently large.