Conjecture B on the averaged binary Goldbach representation count

Let P\mathbb{P} be the set of primes, and let r2(n;P)r_2(n;\mathbb{P}) denote the number of representations of nn as a sum of two primes; let φ\varphi be Euler's totient function. Conjecture B. As nn runs through the even numbers,

r2(n;P)nφ(n)nlog(n)2.r_2(n;\mathbb{P}) \sim \frac{n}{\varphi(n)}\frac{n}{\log(n)^2}.

This conjecture is motivated by an elementary combinatorial heuristic and is intended to approximate the Hardy–Littlewood prediction up to a constant factor. It does not imply the Goldbach conjecture, and the source identifies this asymptotic Goldbach statement as an open problem.

Sources & referencesView supporting material

Primary source

Christian Táfula, “An elementary heuristic for Hardy-Littlewood extended Goldbach's conjecture”, arXiv:1508.05702 (2019).

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