Hardy–Littlewood circle method

Summary

The circle method is an analytic technique for counting the solutions of additive equations. Examples are writing $N$ as a sum of $s$ $k$-th powers, or as a sum of three primes. It writes the number of solutions as an integral over the unit circle of a power of an exponential sum. The integral is split into major arcs near rationals with small denominators, which give the main term. The rest, the minor arcs, are shown to contribute less. It goes back to Hardy and Ramanujan's work on the partition function and was developed by Hardy and Littlewood for Waring's problem.