An asymptotic formula for Goldbach’s conjecture with monic polynomials in $\mathbb {Z}[\theta ][x]$
Colloquium Mathematicum, Tome 148 (2017) no. 2, pp. 215-223
Cet article a éte moissonné depuis la source Institute of Mathematics Polish Academy of Sciences
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 $(4y)^{2d-2}$.
Keywords:
geq squarefree integer theta begin cases sqrt k text k equiv pmod sqrt k text k equiv pmod end cases prove number representations monic polynomial mathbb theta degree geq sum monic irreducible polynomials mathbb theta coefficients bounded modulus asymptotic d
Affiliations des auteurs :
Abílio Lemos 1 ; Anderson Luís Albuquerque de Araujo 1
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author = {Ab{\'\i}lio Lemos and Anderson Lu{\'\i}s Albuquerque de Araujo},
title = {An asymptotic formula for {Goldbach{\textquoteright}s} conjecture with monic polynomials in $\mathbb {Z}[\theta ][x]$},
journal = {Colloquium Mathematicum},
pages = {215--223},
year = {2017},
volume = {148},
number = {2},
doi = {10.4064/cm6948-7-2016},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.4064/cm6948-7-2016/}
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AU - Anderson Luís Albuquerque de Araujo
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Abílio Lemos; Anderson Luís Albuquerque de Araujo. An asymptotic formula for Goldbach’s conjecture with monic polynomials in $\mathbb {Z}[\theta ][x]$. Colloquium Mathematicum, Tome 148 (2017) no. 2, pp. 215-223. doi: 10.4064/cm6948-7-2016
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