On the Piatetski-Shapiro-Vinogradov theorem

Abstract

In this paper we consider the asymptotic formula for the number of the solutions of the equation p1 + p2 + p3 = N where N is an odd integer and the unknowns pi are prime numbers of the form pi = [n1/γi]. We use the two-dimensional van der Corput’s method to prove it under less restrictive conditions than before. In the most interesting case γ1 = γ2 = γ3 = γ our theorem implies that every sufficiently large odd integer N may be written as the sum of three Piatetski-Shapiro primes of type γ for 50/53 < γ < 1.