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 sums.

In finite Galois extensions $k_1,...,k_r$ of $\mathbf{Q}$ with pairwise coprime discriminants the integral and the prime divisors subject to the condition $N_{k_1/\mathbf{Q}}{𝔞}_r=\cdots =N_{k_r/\mathbf{Q}}{𝔞}_r$ are equidistributed in the sense of E. Hecke.

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.

In this paper, we give a new upper-bound for the discrepancy$$D\left(N\right):=\underset{0\le \gamma \le 0}{sup}|\sum _{\genfrac{}{}{0pt}{}{p\le /N}{\left\{p^{\alpha }\right\}\le \gamma }}1-\pi \left(N\right)\gamma |$$for the sequence $\left(p^{\alpha }\right)$, when $5/3\le \alpha \>3$ and $\alpha \ne 2$.

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.