The ternary Goldbach problem with two Piatetski–Shapiro primes and a prime with a missing digit

Abstract

Let [Formula: see text] Let [Formula: see text], [Formula: see text] be fixed. Let also [Formula: see text]. We prove on assumption of the Generalized Riemann Hypothesis that each sufficiently large odd integer [Formula: see text] can be represented in the form [Formula: see text] where the [Formula: see text] are of the form [Formula: see text], [Formula: see text], for [Formula: see text] and the decimal expansion of [Formula: see text] does not contain the digit [Formula: see text]. The proof merges methods of Maynard from his paper on the infinitude of primes with restricted digits, results of Balog and Friedlander on Piatetski-Shapiro primes and the Hardy–Littlewood circle method in two variables. This is the first result on the ternary Goldbach problem with primes of mixed type which involves primes with missing digits.