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| 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]], [[Srinivasa Ramanujan]] and [[J. E. Littlewood]], who developed it in a series of papers on [[Waring's problem]].
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| ==History==
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| 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 (number theory)|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 sum]]s), without changing the broad lines. Hundreds of papers followed, and {{As of|2013|lc=on}} the method still yields results. The method is the subject of a monograph by [[Robert Charles Vaughan (mathematician)|R. C. Vaughan]].
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| ==Outline==
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| The goal is to prove asymptotic behavior of a series: to show that ''a''<sub>''n''</sub> ~ ''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 coefficient]]s). 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.
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| The circle method is specifically how to compute these residues, by [[Partition of an interval|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 function]]s), 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.
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| === Setup ===
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| 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
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| :''a''<sub>''n''</sub>, ''n'' = 0, 1, 2, 3, ... | |
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| we want some asymptotic information of the type
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| :''a''<sub>''n''</sub> ~ ''F''(''n'') | |
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| where we have some [[heuristic]] reason to guess the form taken by ''F'' (an [[ansatz]]), we write
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| :<math>f(z)= \sum a_n z^n </math> | |
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| 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.
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| === Residues ===
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| From that formulation, it is direct from the [[residue theorem]] that
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| :<math>I_n=\int f(z)z^{-(n+1)}\,dz=2 \pi ia_n</math> | |
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| for integers ''n'' ≥ 0, where the integral is taken over the circle of radius ''r'' and centred at 0, for any ''r'' with
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| :0 < ''r'' < 1.
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| That is, this is a [[contour integral]], with the contour being the circle described traversed once anti-clockwise. So far, this is relatively elementary. We would like to take ''r'' = 1 directly, i.e. to use the unit circle contour. In the complex analysis formulation this is problematic, since the values of ''f'' are not in general defined there.
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| === Singularities on unit circle ===
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| 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]]
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| :<math> \zeta\ = \exp \left ( \frac{2 \pi ir}{s} \right ). </math>
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| 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.
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| === Method ===
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| The Hardy–Littlewood circle method, for the complex-analytic formulation, can then be thus expressed. The contributions to the evaluation of ''I''<sub>''n''</sub>, as ''r'' → 1, should be treated in two ways, traditionally called ''major arcs'' and ''minor arcs''. We divide the 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''<sub>''n''</sub> 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 2π''iF''(''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'').
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| == Discussion ==
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| Stated baldly 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 function]]s.
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| === Waring's problem ===
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| In the context of Waring's problem, powers of theta functions are the generating functions for [[sums of squares]]{{dn|date=December 2013}}. Their analytic behaviour is known in much more accurate detail than for the cubes, for example.
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| [[File:Q-Eulero.jpeg|thumb|Typical singular behaviour of a theta function]]
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| 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.
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| 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.
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| === Vinogradov trigonometric sums ===
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| 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''<sub>''n''</sub> is a [[Fourier coefficient]]. Vinogradov applied finite sums to Waring's problem in 1926, and the general trigometric sum method became known as "the circle method of Hardy, Littlewood and Ramanujan, in the form of Vinogradov's trigonometric sums.<ref>Mardzhanishvili (1985), pp. 387–8</ref> 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.
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| == Applications ==
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| Refinements of the method have allowed results to be proved about the solutions of homogeneous [[Diophantine equation]]s, as long as the number of variables ''k'' is large relative to the degree ''d'' (see [[Birch's theorem]] for example). This turns out to be a contribution to the [[Hasse principle]], capable of yielding quantitative information. If ''d'' is fixed and ''k'' is small, other methods are required, and indeed the Hasse principle tends to fail.
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| ==Rademacher's contour==
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| [[File:Circumferències de Ford.svg|450px|right|thumb|[[Ford circles]]: A circle rests upon each fraction in lowest terms. The darker circles shown are for the fractions 0/1, 1/1, 1/2, 1/3, 2/3, 1/4, 3/4, 1/5, 2/5, 3/5 and 4/5. Each circle is [[tangent]]ial to the base line and its neighboring circles (see also [[Tangent lines to circles]]). Fractions with the same denominator have circles of the same size.]]
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| In the special case when the circle method is applied to find the coefficients of a modular form of negative weight, [[Hans Rademacher]] found a modification of the contour that makes the series arising from the circle method converge to the exact result. To describe his contour, it is convenient to replace the unit circle by the upper half plane, by making the substitution ''z'' = exp(2π''i''τ), so that the contour integral becomes an integral from τ = ''i'' to τ = 1 + ''i''. (The number ''i'' could be replaced by any number on the upper half plane, but ''i'' is the most convenient choice.) Rademacher's contour is (more or less) given by the boundaries of all the [[Ford circle]]s from 0 to 1, as shown in the diagram. The replacement of the line from ''i'' to 1 + ''i'' by the boundaries of these circles is a non-trivial limiting process, which can be justified for modular forms that have negative weight, and with more care can also be justified for non-constant terms for the case of weight 0 (in other words [[modular function]]s).
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| ==Notes==
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| {{Reflist}}
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| == References ==
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| *{{Citation | last1=Apostol | first1=Tom M. | author1-link=Tom M. Apostol | title=Modular functions and Dirichlet series in number theory | publisher=[[Springer-Verlag]] | location=Berlin, New York | edition=2nd | isbn=978-0-387-97127-8 | year=1990}}
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| *K. K. Mardzhanishvili'', ''Ivan Matveevich Vinogradov : a brief outline of his life and works, in ''I. M. Vinogradov, Selected works'' (Berlin, 1985)
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| *{{Citation | last1=Rademacher | first1=Hans |authorlink1=Hans Rademacher | title=On the expansion of the partition function in a series | id={{MathSciNet | id = 0008618}} | year=1943 | journal=[[Annals of Mathematics|Annals of Mathematics. Second Series]] | volume=44 | pages=416–422 | doi=10.2307/1968973 | issue=3 | publisher=The Annals of Mathematics, Vol. 44, No. 3 | jstor=1968973}}
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| * {{Citation |last=Vaughan |first=R. C. |authorlink=Robert Charles Vaughan (mathematician) |title=The Hardy–Littlewood Method |edition= 2nd |series=Cambridge Tracts in Mathematics |volume=125 |year=1997 |publisher=Cambridge University Press |isbn=978-0-521-57347-4}}
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| == External links ==
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| *[[Terence Tao]], [http://terrytao.wordpress.com/2012/05/20/heuristic-limitations-of-the-circle-method/#more-5963 Heuristic limitations of the circle method], a blog post in 2012
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| {{DEFAULTSORT:Hardy-Littlewood circle method}}
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| [[Category:Analytic number theory]]
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More aged video games ought not to be discarded. They could be worth some money at several video retailers. After you buy and sell several game titles, you may also get your upcoming bill at no cost!
Obviously if as a parent you're the one concerned with movie recreation content, control what online mods are put previously sport. These online mods are usually created by players, perhaps not your gaming businesses, therefore there is no ranking system. Make use of thought was a reasonably un-risky game can switch off all electronics a lot worse with any of these mods.
Result There are a involving Apple fans who play the above game all worldwide. This generation has no longer been the JRPG's best; in fact it's resulted in being unanimously its worst. Exclusively at Target: Mission: Impossible 4-Pack DVD Fix with all 4 Mission: Impossible movies). If you have any concerns concerning where and how you can make use of clash of clans hack tool (my latest blog post), you could call us at the site. Though it is a special day's grand gifts and gestures, one Valentines Day will probably blend into another too dang easily. clash of clans is among one of the quickest rising video gaming as of late.
It is always possible, but the majority of absence one morning would abatement by 59 one. 5% hailing from 260 treasures to thousand gems. Or, if you capital to construction up the 1 big day bulk at 260 gems, the band would endure to acceleration added considerably and also 1 anniversary would turn into added expensive.
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