Category: Number Theory

Summary: Measuring how quickly the Goldbach partition count approaches the Hardy-Littlewood asymptotic prediction for large even numbers.

The Hardy-Littlewood heuristic gives a detailed asymptotic formula for the number of ways an even integer can be written as a sum of two primes. This experiment asks not only whether the Goldbach partition count follows that prediction on average, but how quickly the ratio approaches its expected limit as numbers grow.

The computation uses sieving to evaluate Goldbach partition counts exactly for large even integers, then compares the results against the asymptotic prediction and candidate correction laws. It also examines the variability of the ratio rather than only its mean trend.

That focus on convergence rate is important because the main asymptotic formula is classical, while finite-size correction behavior is much less sharply characterized. The experiment is designed to map how quickly the number-theoretic data begin to resemble the predicted limit.

Method: Prime-sieving computation of exact Goldbach partition counts followed by scaling analysis of their ratio to the Hardy-Littlewood prediction.

What is measured: Goldbach partition count, ratio to asymptotic prediction, variance of the ratio, and convergence-rate behavior.