View source for Circle method

One of the most general methods in additive number theory. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022270/c0222701.png" /> be arbitrary sets of natural numbers, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022270/c0222702.png" /> be a natural number and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022270/c0222703.png" /> be the number of solutions of the equation

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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022270/c0222705.png" />. It is with the investigation of the numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022270/c0222706.png" /> that additive number theory is concerned; for example, if it can be proved that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022270/c0222707.png" /> is greater than zero for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022270/c0222708.png" />, this means that any natural number is the sum of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022270/c0222709.png" /> terms taken respectively from the sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022270/c02227010.png" />. Now let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022270/c02227011.png" /> be a complex number, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022270/c02227012.png" />, and let

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Then the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022270/c02227014.png" /> defined by

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is the generating function of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022270/c02227016.png" />. By Cauchy's formula,

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The integral in this equality is investigated as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022270/c02227018.png" />. The circle of integration <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022270/c02227019.png" /> is divided into  "major"  and  "minor"  arcs, the centres of which are rational numbers. There is a broad range of additive problems in which the integrals over  "major"  arcs, which yield a  "principal"  part of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022270/c02227020.png" />, can be investigated fairly completely, while the integrals over the  "minor"  arcs, which yield a  "remainder"  term in the asymptotic formula for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022270/c02227021.png" />, can be estimated.

I.M. Vinogradov's use of trigonometric sums in the circle method not only considerably simplified application of the method, it also provided a unified approach to the solution of a wide range of very different additive problems. The basis for the circle method in the form of trigonometric sums is the formula

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It follows from this formula that

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where

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The finite sums <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022270/c02227025.png" /> are called trigonometric sums. To investigate the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022270/c02227026.png" />, one divides the integration interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022270/c02227027.png" /> into  "major"  and  "minor"  arcs, i.e. intervals centred at rational points with  "small"  and  "large"  denominators. For many additive problems one can successfully evaluate — with adequate accuracy — the integrals over the  "major"  arcs (the trigonometric sums for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022270/c02227028.png" /> in  "major"  arcs are close to rational trigonometric sums with small denominators, which are readily evaluated and are  "large" ); as for the  "minor"  arcs, which contain the bulk of the points in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022270/c02227029.png" />, the trigonometric sums over these are  "small" ; they can be estimated in a non-trivial manner (see [[Trigonometric sums, method of|Trigonometric sums, method of]]; [[Vinogradov method|Vinogradov method]]), so that asymptotic formulas can be established for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022270/c02227030.png" />.

The circle method in the trigonometric sum version, together with Vinogradov's method for estimating trigonometric sums, yields the strongest results of additive number theory (see [[Waring problem|Waring problem]]; [[Goldbach problem|Goldbach problem]]; [[Goldbach–Waring problem|Goldbach–Waring problem]]; [[Hilbert–Kamke problem|Hilbert–Kamke problem]]).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  I.M. Vinogradov,   "The method of trigonometric sums in the theory of numbers" , Interscience  (1954)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  L.-K. Hua,   "The method of trigonometric sums and its applications to number theory" , ''Selected papers'' , Springer  (1983)  pp. 124–135  (Translated from German)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A.A. Karatsuba,   "Fundamentals of analytic number theory" , Moscow  (1975)  (In Russian)</TD></TR></table>



====Comments====
The circle method as described above is often referred to as the Hardy–Littlewood method or the Hardy–Littlewood circle method. The method can be adapted to a number of quite diverse situations. Some examples follow. The Davenport–Heilbron theorem says that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022270/c02227031.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022270/c02227032.png" />, are real numbers, not all of the same sign if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022270/c02227033.png" /> is even, and such that at least one ratio <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022270/c02227034.png" /> is irrational, then for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022270/c02227035.png" /> there are integers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022270/c02227036.png" />, not all zero, such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022270/c02227037.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022270/c02227038.png" /> be a subset of the natural numbers such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022270/c02227039.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022270/c02227040.png" /> is the upper [[Asymptotic density|asymptotic density]]. Then the Furstenberg–Sárközy theorem says that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022270/c02227041.png" /> is the number of solutions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022270/c02227042.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022270/c02227043.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022270/c02227044.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022270/c02227045.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022270/c02227046.png" />. Finally there is e.g. Birch's theorem to the effect that the dimension of the space of simultaneous zeros of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022270/c02227047.png" /> homogeneous forms of odd degree grows arbitrarily large with the number of variables of those forms.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R.C. Vaughan,   "The Hardy–Littlewood method" , Cambridge Univ. Press  (1981)</TD></TR></table>