On a numerical upper bound for the extended Goldbach conjecture

Abstract

The Goldbach conjecture states that every even number can be decomposed as the sum of two primes. Let D(N) denote the number of such prime decompositions for an even N. It is known that D(N) can be bounded above by D(N) C N log2 N Y pjN p>2 1 + 1 p ?? 2 Y p>2 1 ?? 1 (p ?? 1)2 = C (N) where C denotes Chen's constant. It is conjectured [20] that C = 2. In 2004,Wu [54] showed that C 7:8209. We attempted to replicate his work in computing Chen's constant, and in doing so we provide an improved approximation of the Buchstab function !(u), !(u) = 1=u Show more