Jinyun Qi

A generalization of Piatetski-Shapiro sequences (II)

Suppose that $α,β\in\mathbb{R}$. Let $α\geqslant1$ and $c$ be a real number in the range $1<c< 12/11$. In this paper, it is proved that there exist infinitely many primes in the generalized Piatetski--Shapiro sequence, which is defined by $(\lfloorαn^c+β\rfloor)_{n=1}^\infty$. Moreover, we also prove that there exist infinitely many Carmichael numbers composed entirely of primes from the generalized Piatetski--Shapiro sequences with $c\in(1,\frac{19137}{18746})$. The two theorems constitute improvements upon previous results by Guo and Qi.