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

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

2010-08-11

arxiv.org/abs/1008.1968v1

Mathematics

Number Theory

Scientific paper

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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