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

arXiv2108.13132MaRDI QIDQ6376324

Helmut Maier, Michael Th. Rassias

Publication date: 30 August 2021

Abstract: Let ${\displaystyle }$gamma^*:=frac{8}{9}+frac{2}{3}:frac{log(10/9)}{log 10}:(approx 0.919ldots):, gamma^*<frac{1}{c_0}leq 1:.${\displaystyle }$ Let $gamm{a}^{*}<gamm{a}_{0}leq1$, $c_0=1/gamm{a}_0$ be fixed. Let also $a_{0}in0,1,ldots,9$. 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 ${\displaystyle }$N_0=p_1+p_2+p_3:,${\displaystyle }$ where for $i=2,3$ the primes $p_i$ are Piatetski-Shapiro primes - primes of the form $p_i=[n_i^{c_0}]$, $n_{i}inmathbbN$ - 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 ${\displaystyle }$p=x^2+y^2+1:.${\displaystyle }$

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