The size of coefficients of certain polynomials related to the Goldbach conjecture

TL;DR

This paper studies the coefficients of polynomials related to the Goldbach conjecture, providing asymptotic formulas for their behavior both conditionally and unconditionally, advancing understanding of their properties.

Contribution

The authors improve previous bounds by deriving an asymptotic formula for the coefficients both conditionally and unconditionally, deepening insight into their structure.

Findings

Asymptotic formula for coefficients under Hardy-Littlewood conjecture

Unconditional asymptotic formula for the summatory function

Enhanced understanding of polynomial coefficients related to Goldbach

Abstract

Recent work of Borwein, Choi, and the second author examined a collection of polynomials closely related to the Goldbach conjecture: the polynomial $F_N$ is divisible by the $N$th cyclotomic polynomial if and only if there is no representation of $N$ as the sum of two odd primes. The coefficients of these polynomials stabilize, as $N$ grows, to a fixed sequence $a(m)$; they derived upper and lower bounds for $a(m)$, and an asymptotic formula for the summatory function $A(M)$ of the sequence, both under the assumption of a famous conjecture of Hardy and Littlewood. In this article we improve these results: we obtain an asymptotic formula for $a(m)$ under the same assumption, and we establish the asymptotic formula for $A(M)$ unconditionally.