The ternary Goldbach problem with a prime with a missing digit and primes of special types

TL;DR

This paper extends previous work on representing large odd integers as sums of primes, incorporating primes with specific digit restrictions and primes of special algebraic forms under the assumption of GRH.

Contribution

It introduces a new result replacing one of the Piatetski-Shapiro primes with primes of the form x^2 + y^2 + 1 in the ternary Goldbach problem under GRH.

Findings

Representation of large odd integers with mixed prime types

Inclusion of primes of the form x^2 + y^2 + 1

Conditional on the Generalized Riemann Hypothesis

Abstract

Let $$\gamma^*:=\frac{8}{9}+\frac{2}{3}\:\frac{\log(10/9)}{\log 10}\:(\approx 0.919\ldots)\:,\ \gamma^*<\frac{1}{c_0}\leq 1\:.$$ Let $\gamma^*<\gamma_0\leq 1$, $c_0=1/\gamma_0$ be fixed. Let also $a_0\in\{0,1,\ldots, 9\}$. In [23] we proved on assumption of the Generalized Riemann Hypothesis (GRH), that each sufficiently large odd integer $N_0$ can be represented in the form $$N_0=p_1+p_2+p_3\:,$$ where for $i=2, 3$ the primes $p_i$ are Piatetski-Shapiro primes - primes of the form $p_i=[n_i^{c_0}]$, $n_i\in\mathbb{N}$ - whereas the decimal expansion of $p_1$ does not contain the digit $a_0$. In this paper we replace one of the Piatetski-Shapiro primes $p_2$ and $p_3$ by primes of the type $$p=x^2+y^2+1\:.$$