An asymptotic formula for Goldbach’s conjecture with monic polynomials in
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}.
