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

TL;DR

Under the assumption of the Generalized Riemann Hypothesis, the paper proves that sufficiently large odd integers can be expressed as a sum of two Piatetski-Shapiro primes and a prime missing a specific digit, combining advanced methods in prime number theory.

Contribution

This work is the first to address the ternary Goldbach problem involving primes of mixed types, including primes with missing digits and Piatetski-Shapiro primes, using a novel combination of existing techniques.

Findings

Proves representation of large odd integers as sum of two Piatetski-Shapiro primes and a missing digit prime.

Extends Goldbach-type results to primes with digit restrictions and special forms.

Utilizes a blend of methods from Maynard, Balog, Friedlander, and Hardy-Littlewood circle method.

Abstract

Let $$\gamma^*=\frac{8}{9}+\frac{2}{3}\:\frac{\log(10/9)}{\log 10}\:(\approx 0.919\ldots)\:.$$ Let $\gamma^*<\gamma_0\leq 1$, $c_0=1/\gamma_0$ be fixed. Let also $a_0\in\{0,1,\ldots, 9\}$.\\ We prove on assumption of the Generalized Riemann Hypothesis that each sufficiently large odd integer $N_0$ can be represented in the form $$N_0=p_1+p_2+p_3\:,$$ where the $p_i$ are of the form $p_i=[n_i^{c_0}]$, $n_i\in\mathbb{N}$, for $i=1,2$ and the decimal expansion of $p_3$ does not contain the digit $a_0$.\\ The proof merges methods of J. Maynard from his paper on the infinitude of primes with restricted digits, results of A. Balog and J. 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.