An asymptotic formula for Goldbach’s conjecture with monic polynomials in $\mathbb {Z}[\theta ][x]$
Volume 148 / 2017
Colloquium Mathematicum 148 (2017), 215-223
MSC: Primary 11R09; Secondary 11C08.
DOI: 10.4064/cm6948-7-2016
Published online: 9 March 2017
Abstract
Let $k\geq 2$ be a squarefree integer, and $$ \theta=\begin{cases} \sqrt{-k} &\text{if }-k\not\equiv 1 \pmod4,\\ {(\sqrt{-k}+1)}/{2} &\text{if }-k\equiv 1 \pmod4.\end{cases} $$ We prove that the number $R(y)$ of representations of a monic polynomial $f(x)\in \mathbb Z[\theta][x]$, of degree $d\geq 1$, as a sum of two monic irreducible polynomials $g(x)$ and $h(x)$ in $\mathbb Z[\theta][x]$, with the coefficients of $g(x)$ and $h(x)$ bounded in modulus by $y$, is asymptotic to $(4y)^{2d-2}$.
