Let N be a set of natural numbers and Z be a set of integers. Let M₂(Z) denotes the set of all 2x2 matrices with integer entries. We give necessary and suficient conditions for solvability of the matrix negative Pell equation (P) X² - dY² = -I with d ∈ N for nonsingular X,Y belonging to M₂(Z) and his generalization (Pn) ${\sum }_{i=1}^{n}X{₂}_i-d{\sum }_{i=1}^{n}Y{²}_i=-I$ with d ∈ N for nonsingular $X_i,Y_i\in M₂\left(Z\right)$, i=1,...,n.

We prove that the sums $S_k$ of independent random vectors satisfy $P(ma{x}_{1\le k\le n}\parallel S_k\parallel >3t)\le 2ma{x}_{1\le k\le n}P(\parallel S_k\parallel >t)$, t ≥ 0.

We obtain a metrical property on the asymptotic behaviour of the maximal run-length function in the Lüroth expansion. We also determine the Hausdorff dimension of a class of exceptional sets of points whose maximal run-length function has sub-linear growth rate.

For the cyclotomic ${\mathbb{Z}}_2$-extension $k_{\infty }$ of an imaginary quadratic field $k$, we consider the Galois group $G\left(k_{\infty }\right)$ of the maximal unramified pro-$2$-extension over $k_{\infty }$. In this paper, we give some families of $k$ for which $G\left(k_{\infty }\right)$ is a metabelian pro-$2$-group with the explicit presentation, and determine the case that $G\left(k_{\infty }\right)$ becomes a nonabelian metacyclic pro-$2$-group. We also calculate Iwasawa theoretically the Galois groups of $2$-class field towers of certain cyclotomic $2$-extensions.

We provide upper bounds for the mean square integral$${\int }_X^{2X}{\left({𝔻}_k(x+h)-{𝔻}_k\left(x\right)\right)}^2\mathrm{d}x,$$where $h=h\left(X\right)\gg 1,h=o\left(x\right)\mathrm{as}X\to \infty$ and $h$ lies in a suitable range. For $k\ge 2$ a fixed integer, ${𝔻}_k\left(x\right)$ is the error term in the asymptotic formula for the summatory function of the divisor function $d_k\left(n\right)$, generated by ${\zeta }^k\left(s\right)$.