21 The Gauss circle problem and its generalizations

This chapter is not yet integrated into the main blueprint.

For any fixed integer \(k \ge 2\) and unbounded \(R\), consider the problem of estimating the number of integer lattice points contained in \(B_k(R)\), a \(k\)-dimensional ball of radius \(R\):

\[ S_k(R) := \# \mathbb {Z}^k \cap B_k(R) = \# \{ x \in \mathbb {Z}^k: |x| \le R\} . \]

Equivalently, \(S_k(R)\) may be written as the partial sum

\[ S_k(R) = \sum _{n \le R^{2}}r_k(n) \]

where \(r_k(n)\) counts the number of integer solutions to the equation \(x_1^2 + \cdots + x_k^2 = n\).

By considering the volume of a \(k\)-dimensional ball of radius \(R\), one has the asymptotic

\[ S_k(R) \sim \operatorname {Vol}(B_k(R)) = \frac{\pi ^{k/2}}{\Gamma (k/2 + 1)}R^k. \]

The generalized Gauss circle problem concerns estimating the error term in this approximation.

Definition 21.1

For fixed integer \(k \ge 2\), define \(\theta ^{\operatorname {Gauss}}_{k}\) as the least (fixed) exponent for which

\[ S_k(R) - \operatorname {Vol}(B_k(R)) \ll R^{\theta ^{\operatorname {Gauss}}_{k} + o(1)}. \]

Figure 21.1 and Figure 21.2 plots the magnitude of this error term for \(k = 2\) and \(k = 3\) respectively (for \(0 {\lt} R \le 1000\)).

\includegraphics[width=0.5\linewidth ]{chapter/gauss_circle_error_2.png} — Figure 21.1 \(|S_k(R) - \operatorname {Vol}(B_k(R))|\) for \(k = 2\) and \(0 {\lt} R \le 1000\)

\includegraphics[width=0.5\linewidth ]{chapter/gauss_circle_error_3.png} — Figure 21.2 \(|S_k(R) - \operatorname {Vol}(B_k(R))|\) for \(k = 3\) and \(0 {\lt} R \le 1000\)

It is conjectured that

Heuristic 21.2

One has

\[ \theta ^{\operatorname {Gauss}}_{k} = \begin{cases} 1/2,& k = 2,\\ k - 2,& k \ge 3. \end{cases} \]

21.1 Known upper and lower bounds

Heuristic 21.2 is known to hold for \(k \ge 4\), i.e.

Theorem 21.3

For integer \(k \ge 4\), one has \(\theta ^{\operatorname {Gauss}}_{k} = k - 2\).

The remaining open cases are \(k = 2, 3\). For such cases the following lower-bounds on \(\theta ^{\operatorname {Gauss}}_{k}\) are known:

Theorem 21.4

One has \(\theta ^{\operatorname {Gauss}}_{2} \ge 1/2\) and \(\theta ^{\operatorname {Gauss}}_{3} \ge 1\).

In light of Heuristic 21.2 and Theorem 21.4, in the rest of this section we shall focus on upper bounds on \(\theta ^{\operatorname {Gauss}}_{k}\) for \(k = 2, 3\).

The case \(k = 2\) is known classically as Gauss’s circle problem. The current sharpest known bound on \(\theta _2^{\operatorname {Gauss}}\) is

Theorem 21.5 Li–Yang (2023) [ 186 ]

One has \(\theta _2^{\operatorname {Gauss}} \le 2\alpha \), where \(\alpha = 0.31448\ldots \) is the solution to the equation

\[ \frac{8}{25}\alpha - \frac{(\sqrt{2(1+14\alpha )} - 5\sqrt{-1+8\alpha })^2}{200} + \frac{51}{200} = \alpha \]

on the interval \([0.3, 0.35]\).

Remark 21.6

The value of \(\alpha \) is the same as that appearing in Theorem 16.7. Historically, methods used to make progress in the \(\alpha _2\) exponent in the Dirichlet divisor problem have led to corresponding improvements in \(\theta _2^{\operatorname {Gauss}}\) (and vice versa). This may be unsurprising given that both problems reduce to counting the number of lattice points contained in a curved region with a smooth boundary (with the region being the hyperbola \(\{ (m,n) \in [0, \infty )^2: mn \le x \} \) in the case of the Dirichlet divisor problem).

The historical progression of upper bounds on \(\theta _2^{\operatorname {Gauss}}\) is recorded in Table 21.1 and Figure 21.3.

Table 21.1 Historical upper bounds on \(\theta _2^{\operatorname {Gauss}}\)

Reference	Bound on \(\theta _2^{\operatorname {Gauss}}\)
Gauss (1834)	\(1\)
Sierpiński (1906) [ 266 ]	\(2/3 = 0.6666\ldots \)
van der Corput (1923) [ 282 ]	\(2/3 - \delta \) for some \(\delta {\gt} 0\)
Littlewood–Walfisz (1924) [ 194 ]	\(37/56 = 0.6607\ldots \)
Walfisz (1927) [ 289 ]	\(163/247 = 0.6599\ldots \)
Nieland (1928) [ 225 ]	\(27/41 = 0.6585\ldots \)
Titchmarsh (1935) [ 274 ]	\(15/23 = 0.6521\ldots \)
Hua (1942) [ 121 ]	\(13/20 = 0.65\)
Iwaniec–Mozzochi (1988) [ 150 ]	\(7/11 = 0.6363\ldots \)
Huxley (1993) [ 127 ]	\(46/73 = 0.6301\ldots \)
Huxley (2003) [ 130 ]	\(131/208 = 0.6298\ldots \)
Li–Yang (2023) [ 186 ]	\(2\alpha * = 0.6289\ldots \)

\includegraphics[width=0.5\linewidth ]{chapter/gauss_circle_historical_bounds.png} — Figure 21.3 Historical upper bounds on \(\theta _2^{\rm Gauss}\)