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arXiv · 1008.1968

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

Greg Martin·Charles L. Samuels
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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.

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Greg Martin, Charles L. Samuels. 2010-08-11. The size of coefficients of certain polynomials related to the Goldbach conjecture. https://arxiv.org/abs/1008.1968

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