View source for Circle problem

The problem of obtaining the best asymptotic estimate for the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022290/c0222901.png" /> of lattice points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022290/c0222902.png" /> in the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022290/c0222903.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022290/c0222904.png" /> be the greatest lower bound of the numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022290/c0222905.png" /> in the equality

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022290/c0222906.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>

C.F. Gauss proved that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022290/c0222907.png" /> (see [[#References|[1]]]). W. Sierpiński, using G.F. Voronoi's method [[#References|[3]]], proved that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022290/c0222908.png" /> (see [[#References|[2]]]). In [[#References|[4]]] it was shown that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022290/c0222909.png" />. The most recent (1987) estimate is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022290/c02229010.png" />. There is a conjecture that the remainder term in formula (*) is

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022290/c02229011.png" /></td> </tr></table>

There is a mean-value theorem for the circle problem:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022290/c02229012.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022290/c02229013.png" /> is some absolute constant and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022290/c02229014.png" /> is arbitrary.

In terms of its content and the methods used to attack it, the circle problem is largely analogous to Dirichlet's divisor problem (see [[Divisor problems|Divisor problems]]). A generalization of the circle problem is the sphere problem — the problem of an estimate for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022290/c02229015.png" />, the number of lattice points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022290/c02229016.png" /> in the ball <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022290/c02229017.png" />. A basis for such an estimate is the formula

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022290/c02229018.png" /></td> </tr></table>

where

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022290/c02229019.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022290/c02229020.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022290/c02229021.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022290/c02229022.png" /></td> </tr></table>

which is established by partitioning the ball into 24 sections by the six planes

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022290/c02229023.png" /></td> </tr></table>

where each section contains the same number of lattice points, provided one counts points on the sectioning planes with coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022290/c02229024.png" />. The principal term of growth of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022290/c02229025.png" /> is equal to the volume of the ball

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022290/c02229026.png" /></td> </tr></table>

hence the problem reduces to estimating <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022290/c02229027.png" />, which is the sum of the fractional parts of the functions in square brackets in the formula for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022290/c02229028.png" />. The most profound estimates for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022290/c02229029.png" /> have been obtained using I.M. Vinogradov's method of trigonometric sums [[#References|[5]]], [[#References|[6]]]:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022290/c02229030.png" /></td> </tr></table>

There is a conjecture that

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022290/c02229031.png" /></td> </tr></table>

A generalization of the circle and sphere problems is the problem of estimating <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022290/c02229032.png" />, the number of lattice points in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022290/c02229034.png" />-dimensional ellipsoids

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022290/c02229035.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022290/c02229036.png" /> is a positive-definite quadratic form [[#References|[7]]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  C.F. Gauss,  , ''Werke'' , '''2''' , Göttingen  (1863)  pp. 269–291</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  W. Sierpiński,   ''Prace Mat. Fiz.'' , '''17'''  (1906)  pp. 77–118</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  G.F. Voronoi,  , ''Collected works'' , '''1''' , Kiev  (1952)  pp. 5  (In Russian)</TD></TR><TR><TD valign="top">[4]</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">[5]</TD> <TD valign="top">  I.M. Vinogradov,   "On the number of integer points in a ball"  ''Izv. Akad. Nauk SSSR Ser. Mat.'' , '''27''' :  5  (1963)  pp. 957–968  (In Russian)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  I.M. Vinogradov,   "Basic variants of the method of trigonometric sums" , Moscow  (1976)  (In Russian)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  B. Novák,   "Lattice points in more-dimensional ellipsoids"  ''Trudy Mat. Inst. Akad. Nauk SSSR'' , '''132'''  (1973)  pp. 145–150  (In Russian)</TD></TR></table>



====Comments====
The most recent estimate referred to above, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022290/c02229037.png" />, was obtained in 1963 by J. Chen [[#References|[a1]]]. The circle problem is also known as the Gauss circle problem.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J. Chen,   "The lattice-points in a circle"  ''Sci. Sinica'' , '''12'''  (1963)  pp. 633–649</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  A.Z. Walfisz,   "Gitterpunkte in mehrdimensionalen Kugeln" , PWN  (1957)</TD></TR></table>