In mathematics, the Hardy–Littlewood circle method is a technique of analytic number theory. It is named for G. H. Hardy and J. E. Littlewood, who developed it in a series of papers on Waring's problem.
History 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. Hundreds of papers followed, and as of 2022 the method still yields results. The method is the subject of a monograph Vaughan (1997) by R. C. Vaughan.
Outline The goal is to prove asymptotic behavior of a series: to show that an ~ F(n) 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 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
f ( z ) = ∑ a n z n {\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
I n = ∮ C f ( z ) z − ( n + 1 ) d z = 2 π i a n {\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, I n {\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:
ζ = exp ( 2 π i r s ) . {\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 In, 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 In 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).
Discussion Stated boldly like this, it is not at all clear that this can be made to work. The insights involved are quite deep. One clear source is the theory of theta functions.
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.
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