Setup

The circle in question was initially the unit circle in the complex plane. Assuming the problem had first been formulated in the terms that for a sequence of complex numbers a_n for n = 0, 1, 2, 3, ..., we want some asymptotic information of the type a_n ~ F(n), where we have some heuristic reason to guess the form taken by F (an ansatz), we write

${\displaystyle f(z)=\sum a_{n}z^{n}}$

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.

Residues

From that formulation, it follows directly from the residue theorem that

${\displaystyle I_{n}=\oint _{C}f(z)z^{-(n+1)}\,dz=2\pi ia_{n}}$

for integers n ≥ 0, where C is a circle of radius r and centred at 0, for any r with 0 < r < 1; in other words, ${\displaystyle I_{n}}$ is a contour integral, integrated over the circle described traversed once anticlockwise. We would like to take r = 1 directly, that is, to use the unit circle contour. In the complex analysis formulation this is problematic, since the values of f may not be defined there.

Singularities on unit circle

The problem addressed by the circle method is to force the issue of taking r = 1, by a good understanding of the nature of the singularities f exhibits on the unit circle. The fundamental insight is the role played by the Farey sequence of rational numbers, or equivalently by the roots of unity:

${\displaystyle \zeta \ =\exp \left({\frac {2\pi ir}{s}}\right).}$

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.

Method

The Hardy–Littlewood circle method, for the complex-analytic formulation, can then be thus expressed. The contributions to the evaluation of I_n, as r → 1, should be treated in two ways, traditionally called major arcs and minor arcs. We divide the roots of unity ζ into two classes, according to whether s ≤ N or s > N, where N is a function of n that is ours to choose conveniently. The integral I_n is divided up into integrals each on some arc of the circle that is adjacent to ζ, of length a function of s (again, at our discretion). The arcs make up the whole circle; the sum of the integrals over the major arcs is to make up 2πiF(n) (realistically, this will happen up to a manageable remainder term). The sum of the integrals over the minor arcs is to be replaced by an upper bound, smaller in order than F(n).

Waring's problem

In the context of Waring's problem, powers of theta functions are the generating functions for the sum of squares function. Their analytic behaviour is known in much more accurate detail than for the cubes, for example.

Typical singular behaviour of a theta function.

It is the case, as the false-colour diagram indicates, that for a theta function the 'most important' point on the boundary circle is at z = 1; followed by z = −1, and then the two complex cube roots of unity at 7 o'clock and 11 o'clock. After that it is the fourth roots of unity i and −i that matter most. 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.

In the case of Waring's problem, one takes a sufficiently high power of the generating function to force the situation in which the singularities, organised into the so-called singular series, predominate. The less wasteful the estimates used on the rest, the finer the results. As Bryan Birch has put it, the method is inherently wasteful. 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.

Vinogradov trigonometric sums

Later, I. M. Vinogradov extended the technique, replacing the exponential sum formulation f(z) with a finite Fourier series, so that the relevant integral I_n is a Fourier coefficient. Vinogradov applied finite sums to Waring's problem in 1926, and the general trigonometric sum method became known as "the circle method of Hardy, Littlewood and Ramanujan, in the form of Vinogradov's trigonometric sums".^[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.