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The Hardy–Littlewood Circle Method

I found this interesting Wikipedia article:

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.


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.


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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