### On the Piatetski-Shapiro-Vinogradov theorem

In this paper we consider the asymptotic formula for the number of the solutions of the equation ${p}_{1}+{p}_{2}+{p}_{3}=N$ where $N$ is an odd integer and the unknowns ${p}_{i}$ are prime numbers of the form ${p}_{i}=\left[{n}^{1/{\gamma}_{i}}\right]$. We use the two-dimensional van der Corput’s method to prove it under less restrictive conditions than before. In the most interesting case ${\gamma}_{1}={\gamma}_{2}={\gamma}_{3}=\gamma $ our theorem implies that every sufficiently large odd integer $N$ may be written as the sum of three Piatetski-Shapiro primes of type $\gamma $ for $50/53$ < $\gamma $ < $1$.