In mathematics, the Hardy–Littlewood circle method is

one of the most frequently used techniques 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 germ of the 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 2005 the method still yields results. The method is the

subject of a monograph by R. C. Vaughan.

Outline

The goal is to prove asymptotic behavior of a series: to show

that a_{n} ~ 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 major arcs (the bulk of the circle) and minor 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.**

Some more interesting articles to follow.

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