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.

Hawkins introduced a probabilistic version of Erathosthenes’ sieve and studied the associated sequence of random “primes” ${\left(p_k\right)}_{k\ge 1}$. Using various probabilistic techniques, many authors have obtained sharp results concerning these random “primes”, which are often in agreement with certain classical theorems or conjectures for prime numbers. In this paper, we prove that the number of integers $k\le n$ such that $p_{k+\alpha }-p_k=\alpha$ is almost surely equivalent to $n/log{\left(n\right)}^{\alpha }$, for a given fixed integer $\alpha \ge 1$. This is a particular case of a recent...

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 $\alpha \ne 2$.