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Recent work of Borwein, Choi, and the second author examined a collection ofpolynomials closely related to the Goldbach conjecture: the polynomial $F_N$ isdivisible by the $N$th cyclotomic polynomial if and only if there is norepresentation of $N$ as the sum of two odd primes. The coefficients of thesepolynomials stabilize, as $N$ grows, to a fixed sequence $a(m)$; they derivedupper and lower bounds for $a(m)$, and an asymptotic formula for the summatoryfunction $A(M)$ of the sequence, both under the assumption of a famousconjecture of Hardy and Littlewood. In this article we improve these results:we obtain an asymptotic formula for $a(m)$ under the same assumption, and weestablish the asymptotic formula for $A(M)$ unconditionally.

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