Circle Method

The background to the invention of the circle method is given, with a description of the Hardy–Ramanujan asymptotic series for the partition function, and the eventual solution in the form of the Rademacher series. Waring’s problem on sums of powers, particularly of cubes, and the problem of the determination, or estimation, of the numbers g(k) and G(k) are considered. There is a proof of the Hardy-Littlewood asymptotic formula of the number of representation of a large numbers as a sum of \(s \ge {2^{k}+1}\) lots of k-th powers, with the coefficient of the main term involving the singular series. The third example of the circle method is Vinogradov’s theorem which solves the ternary Goldbach conjecture on the representation of odd numbers as sums of three primes.