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The problem of obtaining the best asymptotic estimate for the number  $  A ( x) $
of lattice points  $  ( u, v) $
in the disc  $  u  ^ {2} + v  ^ {2} \leq  x $. 
Let  $  \theta $
be the greatest lower bound of the numbers  $  \alpha $
in the equality

$$ \tag{* }
A ( x)  = \ 
\pi x + O ( x  ^  \alpha  ).
$$

C.F. Gauss proved that  $  \theta \leq  1/2 $(
see [[#References|[1]]]). W. Sierpiński, using G.F. Voronoi's method [[#References|[3]]], proved that  $  \theta \leq  1/3 $(
see [[#References|[2]]]). In [[#References|[4]]] it was shown that  $  \theta \leq  13/40 $. 
The most recent (1987) estimate is  $  \theta \leq  12/37 $. 
There is a conjecture that the remainder term in formula (*) is

$$ 
O ( x  ^ {1/4}   \mathop{\rm log}  ^ {2}  x).
$$

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

$$ 
\int\limits _ { 0 } ^ { N } 
( A ( x) - \pi x)  ^ {2}  dx  = \ 
CN  ^ {3/2} + O ( N ^ {1 + \epsilon } ),
$$

where  $  C $
is some absolute constant and  $  \epsilon > 0 $
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  $  B ( x) $, 
the number of lattice points  $  ( u, v, w) $
in the ball  $  u  ^ {2} + v  ^ {2} + w  ^ {2} \leq  x $. 
A basis for such an estimate is the formula

$$ 
B ( x)  = \ 
24G ( x) +
O ( \sqrt x ),
$$

where

$$ 
G ( x) =
$$

$$ 
= \ 
\sum _ {0 < u \leq  \sqrt x/3 }  \sum _ {u < v \leq  \sqrt {( x -
u  ^ {2} )/2 } } ([ \sqrt {x - u  ^ {2} - v  ^ {2} } ] - v) +
$$

$$ 
+ 
{
\frac{1}{2}
 } \sum _ {0 < v \leq  \sqrt x/2 } ([ \sqrt {x - v  ^ {2} } ] - v) +
$$

$$ 
+ 
{
\frac{1}{2}
 } \sum _ {0 < u \leq  \sqrt x/3 } ([ \sqrt
{x - 2u  ^ {2} } ] - u) + O ( \sqrt x ),
$$

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

$$ 
u  =  v,\ \ 
u  =  w,\ \ 
v  =  w,\ \ 
v  =  0,\ \ 
u  =  0,\ \ 
w  =  0,
$$

where each section contains the same number of lattice points, provided one counts points on the sectioning planes with coefficients  $  1/2 $. 
The principal term of growth of  $  B ( x) $
is equal to the volume of the ball

$$ 
V ( x)  = \ 

\frac{4 \pi }{3}

x  ^ {3/2} ;
$$

hence the problem reduces to estimating  $  P ( x) = B ( x) - V ( x) $, 
which is the sum of the fractional parts of the functions in square brackets in the formula for  $  G ( x) $. 
The most profound estimates for  $  P ( x) $
have been obtained using I.M. Vinogradov's method of trigonometric sums [[#References|[5]]], [[#References|[6]]]:

$$ 
P ( x)  = \ 
O ( x  ^ {2/3}   \mathop{\rm log}  ^ {6}  x).
$$

There is a conjecture that

$$ 
P ( x)  = \ 
O ( x  ^ {1/2}   \mathop{\rm log}  ^ {2}  x).
$$

A generalization of the circle and sphere problems is the problem of estimating  $  A _ {F} ( x) $, 
the number of lattice points in  $  n $-
dimensional ellipsoids

$$ 
F ( u _ {1} \dots u _ {n} )  = \ 
\sum _ {r, \nu = 1 } ^ { n } 
a _ {r \nu }  u _ {r} u _  \nu   \leq   x,\ \ 
a _ {r \nu }  = a _ {\nu r }  ,
$$

where  $  F $
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,  $  \theta \leq  12/37 $, 
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>