Let ${𝒫}_r$ denote an almost-prime with at most $r$ prime factors, counted according to multiplicity. Suppose that $a$ and $q$ are positive integers satisfying $(a,q)=1$. Denote by ${𝒫}_2(a,q)$ the least almost-prime ${𝒫}_2$ which satisfies ${𝒫}_2\equiv a(modq)$. It is proved that for sufficiently large $q$, there holds $${𝒫}_2(a,q)\ll q^{1.8345}.$$ This result constitutes an improvement upon that of Iwaniec (1982), who obtained the same conclusion, but for the range $1.845$ in place of $1.8345$.

Let ${𝒫}_2$ denote a positive integer with at most $2$ prime factors, counted according to multiplicity. For integers $a$, $q$ such that $(a,q)=1$, let ${𝒫}_2(q,a)$ denote the least ${𝒫}_2$ in the arithmetic progression ${\{nq+a\}}_{n=1}^{\infty }$. It is proved that for sufficiently large $q$, we have $${𝒫}_2(q,a)\ll q^{1.825}.$$ This result constitutes an improvement upon that of J. Li, M. Zhang and Y. Cai (2023), who obtained ${𝒫}_2(q,a)\ll q^{1.8345}.$

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 ${\zeta }^k\left(s\right)$.