This paper gives further results about the distribution in the arithmetic progressions (modulo a product of two primes) of reducible quadratic polynomials $(an+b)(cn+d)$ in short intervals $n\in [x,x+x^{\vartheta }]$, where now $\vartheta \in (0,1]$. Here we use the Dispersion Method instead of the Large Sieve to get results beyond the classical level $\vartheta$, reaching $3\vartheta /2$ (thus improving also the level of the previous paper, i.e. $3\vartheta -3/2$), but our new results are different in structure. Then, we make a graphical comparison of the two methods.

Let $g_j\left(m\right),j=1,2$, be complex-valued multiplicative functions. In the paper the necessary and sufficient conditions are indicated for the convergence in some sense of probability measure$$\frac{1}{n}\text{card}\left\{0\le m\le n:(g_1\left(m\right),g_2\left(m\right))\in A\right\},A\in ℬ\left({ℂ}^2\right),$$as $n\to \infty$.

A natural number $n$ is said to be a $(k,r)$-integer if $n=a^{k}b$, where $k>r>1$ and $b$ is not divisible by the $r$th power of any prime. We study the distribution of such $(k,r)$-integers in the Piatetski-Shapiro sequence $\left\{\lfloor n^c\rfloor \right\}$ with $c>1$. As a corollary, we also obtain similar results for semi-$r$-free integers.

We give asymptotic formulas for some average values of the Euler function on shifted smooth numbers. The result is based on various estimates on the distribution of smooth numbers in arithmetic progressions which are due to A. Granville and É. Fouvry & G. Tenenbaum.

Let $\left[x\right]$ be an integer part of $x$ and $d\left(n\right)$ be the number of positive divisor of $n$. Inspired by some results of M. Jutila (1987), we prove that for $1<c<\frac{6}{5}$, $$\sum _{n\le x}d\left(\left[n^c\right]\right)=cxlogx+(2\gamma -c)x+O\left(\frac{x}{logx}\right),$$ where $\gamma$ is the Euler constant and $\left[n^c\right]$ is the Piatetski-Shapiro sequence. This gives an improvement upon the classical result of this problem.