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

2010/08/11 by Greg Martin, Martin, Greg, Charles L. Samuels +1
Computer Science · Mathematics · #11A41 #11N37 #Advanced Mathematical Identities #Analytic Number Theory Research #Coding theory and cryptography #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1008.1968

openalex publication_date 2010/08/11 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

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 Nth 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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