A positive integer $n$ is called a square-free number if it is not divisible by a perfect square except $1$. Let $p$ be an odd prime. For $n$ with $(n,p)=1$, the smallest positive integer $f$ such that $n^f\equiv 1(modp)$ is called the exponent of $n$ modulo $p$. If the exponent of $n$ modulo $p$ is $p-1$, then $n$ is called a primitive root mod $p$. Let $A\left(n\right)$ be the characteristic function of the square-free primitive roots modulo $p$. In this paper we study the distribution $$\sum _{n\le x}A\left(n\right)A(n+1),$$ and give an asymptotic formula by using properties of character...

We compare the growth of the least common multiple of the numbers $u_{a_1},...,u_{a_n}$ and $|u_{a_1}\cdots u_{a_n}|$, where ${\left(u_n\right)}_{n\ge 0}$ is a Lucas sequence and ${\left(a_n\right)}_{n\ge 0}$ is some sequence of positive integers.

Let $G$ be a simple algebraic group over an algebraically closed field $𝕜$ of characteristic 0, and $𝔤=LieG$. Let $(e,h,f)$ be an $𝔰{𝔩}_2$-triple in $𝔤$ with $e$ being a long root vector in $𝔤$. Let $(·,·)$ be the $G$-invariant bilinear form on $𝔤$ with $(e,f)=1$ and let $\chi \in {𝔤}^{*}$ be such that $\chi \left(x\right)=(e,x)$ for all $x\in 𝔤$. Let $𝒮$ be the Slodowy slice at $e$ through the adjoint orbit of $e$ and let $H$ be the enveloping algebra of $𝒮$; see [31]. In this article we give an explicit presentation of $H$ by generators and relations. As a consequence we deduce that $H$ contains...

A metacyclic group $H$ can be presented as $\langle \alpha ,\beta :{\alpha }^n=1$, ${\beta }^m={\alpha }^t$, $\beta \alpha {\beta }^{-1}={\alpha }^r\rangle$ for some $n$, $m$, $t$, $r$. Each endomorphism $\sigma$ of $H$ is determined by $\sigma \left(\alpha \right)={\alpha }^{x_1}{\beta }^{y_1}$, $\sigma \left(\beta \right)={\alpha }^{x_2}{\beta }^{y_2}$ for some integers $x_1$, $x_2$, $y_1$, $y_2$. We give sufficient and necessary conditions on $x_1$, $x_2$, $y_1$, $y_2$ for $\sigma$ to be an automorphism.

Let $X\subset {\mathbb{P}}^n$ be an integral and non-degenerate $m$-dimensional variety defined over $ℝ$. For any $P\in {\mathbb{P}}^n\left(ℝ\right)$ the real $X$-rank $r_{X,ℝ}\left(P\right)$ is the minimal cardinality of $S\subset X\left(ℝ\right)$ such that $P\in \langle S\rangle$. Here we extend to the real case an upper bound for the $X$-rank due to Landsberg and Teitler.

Let $R$ be a commutative ring, $M$ an $R$-module and $G$ a group of $R$-automorphisms of $M$, usually with some sort of rank restriction on $G$. We study the transfer of hypotheses between $M/C_M\left(G\right)$ and $[M,G]$ such as Noetherian or having finite composition length. In this we extend recent work of Dixon, Kurdachenko and Otal and of Kurdachenko, Subbotin and Chupordia. For example, suppose $[M,G]$ is $R$-Noetherian. If $G$ has finite rank, then $M/C_M\left(G\right)$ also is $R$-Noetherian. Further, if $[M,G]$ is $R$-Noetherian and if only certain abelian...

Let $p=1+2^{e+1}q$ be an odd prime number with q an odd integer. Let δ (resp. φ) be an odd (resp. even) Dirichlet character of conductor p and order $2^{e+1}$ (resp. order $d_{\phi }$ dividing q), and let ψₙ be an even character of conductor $p^{n+1}$ and order pⁿ. We put χ = δφψₙ, whose value is contained in $Kₙ=\mathbb{Q}\left({\zeta }_{(p-1)pⁿ}\right)$. It is well known that the Bernoulli number $B_{1,\chi }$ is not zero, which is shown in an analytic way. In the extreme cases $d_{\phi }=1$ and q, we show, in an algebraic and elementary manner, a stronger nonvanishing result: $T{r}_{n/1}\left(\xi B_{1,\chi }\right)\ne 0$ for any...

A graph $X$, with a group $G$ of automorphisms of $X$, is said to be $(G,s)$-transitive, for some $s\ge 1$, if $G$ is transitive on $s$-arcs but not on $(s+1)$-arcs. Let $X$ be a connected $(G,s)$-transitive graph of prime valency $p\ge 5$, and $G_v$ the vertex stabilizer of a vertex $v\in V\left(X\right)$. Suppose that $G_v$ is solvable. Weiss (1974) proved that $|G_v{|\mid p(p-1)}^2$. In this paper, we prove that $G_v\cong ({\mathbb{Z}}_p⋊{\mathbb{Z}}_m)\times {\mathbb{Z}}_n$ for some positive integers $m$ and $n$ such that $ndivm$ and $m\mid p-1$.

Let $D$ be a ${𝒞}^d$ $q$-convex intersection, $d\ge 2$, $0\le q\le n-1$, in a complex manifold $X$ of complex dimension $n$, $n\ge 2$, and let $E$ be a holomorphic vector bundle of rank $N$ over $X$. In this paper, ${𝒞}^k$-estimates, $k=2,3,\cdots ,\infty$, for solutions to the $\overline{\partial }$-equation with small loss of smoothness are obtained for $E$-valued $(0,s)$-forms on $D$ when $n-q\le s\le n$. In addition, we solve the $\overline{\partial }$-equation with a support condition in ${𝒞}^k$-spaces. More precisely, we prove that for a $\overline{\partial }$-closed form $f$ in ${𝒞}_{0,q}^k(X\setminus D,E)$, $1\le q\le n-2$, $n\ge 3$, with compact support and for $\epsilon$ with $0<\epsilon <1$ there...

Let $G$ be a group and $\omega \left(G\right)$ be the set of element orders of $G$. Let $k\in \omega \left(G\right)$ and $m_k\left(G\right)$ be the number of elements of order $k$ in $G$. Let nse$\left(G\right)=\{m_k\left(G\right):k\in \omega \left(G\right)\}$. Assume $r$ is a prime number and let $G$ be a group such that nse$\left(G\right)=$ nse$\left(S_r\right)$, where $S_r$ is the symmetric group of degree $r$. In this paper we prove that $G\cong S_r$, if $r$ divides the order of $G$ and $r^2$ does not divide it. To get the conclusion we make use of some well-known results on the prime graphs of finite simple groups and their components.