The goal is to prove asymptotic behavior of a series: to show that for some function. This is done by taking the
generating function of the series, then computing the
residues about zero (essentially the
Fourier coefficients). Technically, the generating function is scaled to have radius of convergence 1, so it has singularities on the unit circle – thus one cannot take the contour integral over the unit circle. The circle method is specifically how to compute these residues, by
partitioning the circle into minor arcs (the bulk of the circle) and major arcs (small arcs containing the most significant singularities), and then bounding the behavior on the minor arcs. The key insight is that, in many cases of interest (such as
theta functions), the singularities occur at the
roots of unity, and the significance of the singularities is in the order of the
Farey sequence. Thus one can investigate the most significant singularities, and, if fortunate, compute the integrals.
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 for , we want some asymptotic information of the type , where we have some
heuristic reason to guess the form taken by (an
ansatz), we write :f(z)= \sum a_n z^n a
power series generating function. The interesting cases are where 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 :I_n=\oint_{C} f(z)z^{-(n+1)}\,dz = 2\pi ia_n for integers , where is a circle of radius and centred at 0, for any with ; in other words, I_n is a
contour integral, integrated over the circle described traversed once anticlockwise. We would like to take directly, that is, to use the unit circle contour. In the complex analysis formulation this is problematic, since the values of may not be defined there.
Singularities on unit circle The problem addressed by the circle method is to force the issue of taking , 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: : \zeta\ = \exp \left ( \frac{2 \pi ir}{s} \right ). Here the
denominator , assuming that is
in lowest terms, turns out to determine the relative importance of the singular behaviour of typical near .
Method The Hardy–Littlewood circle method, for the complex-analytic formulation, can then be thus expressed. The contributions to the evaluation of , as , should be treated in two ways, traditionally called
major arcs and
minor arcs. We divide the roots of unity into two classes, according to whether or , where is a function of that is ours to choose conveniently. The integral is divided up into integrals each on some arc of the circle that is adjacent to , of length a function of (again, at our discretion). The arcs make up the whole circle; the sum of the integrals over the
major arcs is to make up (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 . == Discussion ==