Circle problem

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 [1]). W. Sierpiński, using G.F. Voronoi's method [3], proved that $ \theta \leq 1/3 $( see [2]). In [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). 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 [5], [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 [7].

References

[1]	C.F. Gauss, , Werke , 2 , Göttingen (1863) pp. 269–291
[2]	W. Sierpiński, Prace Mat. Fiz. , 17 (1906) pp. 77–118
[3]	G.F. Voronoi, , Collected works , 1 , Kiev (1952) pp. 5 (In Russian)
[4]	L.-K. Hua, "The method of trigonometric sums and its applications to number theory" , Selected papers , Springer (1983) pp. 124–135 (Translated from German)
[5]	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)
[6]	I.M. Vinogradov, "Basic variants of the method of trigonometric sums" , Moscow (1976) (In Russian)
[7]	B. Novák, "Lattice points in more-dimensional ellipsoids" Trudy Mat. Inst. Akad. Nauk SSSR , 132 (1973) pp. 145–150 (In Russian)

Comments

The most recent estimate referred to above, $ \theta \leq 12/37 $, was obtained in 1963 by J. Chen [a1]. The circle problem is also known as the Gauss circle problem.

References

[a1]	J. Chen, "The lattice-points in a circle" Sci. Sinica , 12 (1963) pp. 633–649
[a2]	A.Z. Walfisz, "Gitterpunkte in mehrdimensionalen Kugeln" , PWN (1957)