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...

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 $\alpha \ne 2$.

We consider an approximation to the popular conjecture about representations of integers as sums of four squares of prime numbers.

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 ${𝒫}_2(q,a)\ll q^{1.8345}.$