The Piatetski-Shapiro prime number theorem

The Piatetski-Shapiro sequences are of the form ${\mathcal{N}}_c:=(\lfloor n^c\rfloor {)}_{n=1}^{\mathrm{\infty }}$, where $\lfloor \cdot \rfloor$ is the integer part. It is expected that there are infinitely many primes in a Piatetski-Shapiro sequence for $c\in (1,2)$. In this article, we prove there are infinitely many Piatetski-Shapiro prime numbers for $1<c<1.1612\dots$ with an asymptotic formula. As a key idea, we prove a new bound for related type $I$ sum.