Hardy–Ramanujan–Littlewood circle method

The initial idea is usually attributed to the work of Hardy with Srinivasa Ramanujan a few years earlier, in 1916 and 1917, on the asymptotics of the partition function.

It was taken up by many other researchers, including Harold Davenport and I. M. Vinogradov, who modified the formulation slightly (moving from complex analysis to exponential sums), without changing the broad lines.

Assuming the problem had first been formulated in the terms that for a sequence of complex numbers an for n = 0, 1, 2, 3, ..., we want some asymptotic information of the type an ~ F(n), where we have some heuristic reason to guess the form taken by F (an ansatz), we write a power series generating function.

The interesting cases are where f is then of radius of convergence equal to 1, and we suppose that the problem as posed has been modified to present this situation.

The fundamental insight is the role played by the Farey sequence of rational numbers, or equivalently by the roots of unity: Here the denominator s, assuming that ⁠r/s⁠ is in lowest terms, turns out to determine the relative importance of the singular behaviour of typical f near ζ.

While nothing in this guarantees that the analytical method will work, it does explain the rationale of using a Farey series-type criterion on roots of unity.

That does not apply to the case of the partition function, which signalled the possibility that in a favourable situation the losses from estimates could be controlled.

[1] Essentially all this does is to discard the whole 'tail' of the generating function, allowing the business of r in the limiting operation to be set directly to the value 1.

Refinements of the method have allowed results to be proved about the solutions of homogeneous Diophantine equations, as long as the number of variables k is large relative to the degree d (see Birch's theorem for example).

In the special case when the circle method is applied to find the coefficients of a modular form of negative weight, Hans Rademacher found a modification of the contour that makes the series arising from the circle method converge to the exact result.