Sum of squares function

In number theory, the sum of squares function is an arithmetic function that gives the number of representations for a given positive integer n as the sum of k squares, where representations that differ only in the order of the summands or in the signs of the numbers being squared are counted as different, and is denoted by r_k(n).

Definition

The function is defined as

[math]\displaystyle{ r_k(n) = |\{(a_1, a_2, \ldots, a_k) \in \mathbb{Z}^k \ : \ n = a_1^2 + a_2^2 + \cdots + a_k^2\}| }[/math]

where [math]\displaystyle{ |\,\ | }[/math] denotes the cardinality of a set. In other words, r_k(n) is the number of ways n can be written as a sum of k squares.

For example, [math]\displaystyle{ r_2(1) = 4 }[/math] since [math]\displaystyle{ 1 = 0^2 + (\pm 1)^2 = (\pm 1)^2 + 0^2 }[/math] where each sum has two sign combinations, and also [math]\displaystyle{ r_2(2) = 4 }[/math] since [math]\displaystyle{ 2 = (\pm 1)^2 + (\pm 1)^2 }[/math] with four sign combinations. On the other hand, [math]\displaystyle{ r_2(3) = 0 }[/math] because there is no way to represent 3 as a sum of two squares.

Formulae

k = 2

Integers satisfying the sum of two squares theorem are squares of possible distances between integer lattice points; values up to 100 are shown, with

•	Squares (and thus integer distances) in red
•	Non-unique representations (up to rotation and reflection) bolded

The number of ways to write a natural number as sum of two squares is given by r₂(n). It is given explicitly by

[math]\displaystyle{ r_2(n) = 4(d_1(n)-d_3(n)) }[/math]

where d₁(n) is the number of divisors of n which are congruent to 1 modulo 4 and d₃(n) is the number of divisors of n which are congruent to 3 modulo 4. Using sums, the expression can be written as:

[math]\displaystyle{ r_2(n) = 4\sum_{d \mid n \atop d\,\equiv\,1,3 \pmod 4}(-1)^{(d-1)/2} }[/math]

The prime factorization [math]\displaystyle{ n = 2^g p_1^{f_1}p_2^{f_2}\cdots q_1^{h_1}q_2^{h_2}\cdots }[/math], where [math]\displaystyle{ p_i }[/math] are the prime factors of the form [math]\displaystyle{ p_i \equiv 1\pmod 4, }[/math] and [math]\displaystyle{ q_i }[/math] are the prime factors of the form [math]\displaystyle{ q_i \equiv 3\pmod 4 }[/math] gives another formula

[math]\displaystyle{ r_2(n) = 4 (f_1 +1)(f_2+1)\cdots }[/math], if all exponents [math]\displaystyle{ h_1, h_2, \cdots }[/math] are even. If one or more [math]\displaystyle{ h_i }[/math] are odd, then [math]\displaystyle{ r_2(n) = 0 }[/math].

k = 3

Gauss proved that for a squarefree number n > 4,

[math]\displaystyle{ r_3(n) = \begin{cases} 24 h(-n), & \text{if } n\equiv 3\pmod{8}, \\ 0 & \text{if } n\equiv 7\pmod{8}, \\ 12 h(-4n) & \text{otherwise}, \end{cases} }[/math]

where h(m) denotes the class number of an integer m.

There exist extensions of Gauss' formula to arbitrary integer n.

k = 4

The number of ways to represent n as the sum of four squares was due to Carl Gustav Jakob Jacobi and it is eight times the sum of all its divisors which are not divisible by 4, i.e.

[math]\displaystyle{ r_4(n)=8\sum_{d\,\mid\,n,\ 4\,\nmid\,d}d. }[/math]

Representing n = 2^km, where m is an odd integer, one can express [math]\displaystyle{ r_4(n) }[/math] in terms of the divisor function as follows:

[math]\displaystyle{ r_4(n) = 8\sigma(2^{\min\{k,1\}}m). }[/math]

k = 6

The number of ways to represent n as the sum of six squares is given by

[math]\displaystyle{ r_6(n) = 4\sum_{d\mid n} d^2\big( 4\left(\tfrac{-4}{n/d}\right) - \left(\tfrac{-4}{d}\right)\big), }[/math]

where [math]\displaystyle{ \left(\tfrac{\cdot}{\cdot}\right) }[/math] is the Kronecker symbol.

k = 8

Jacobi also found an explicit formula for the case k = 8:

[math]\displaystyle{ r_8(n) = 16\sum_{d\,\mid\,n}(-1)^{n+d}d^3. }[/math]

Generating function

The generating function of the sequence [math]\displaystyle{ r_k(n) }[/math] for fixed k can be expressed in terms of the Jacobi theta function:

[math]\displaystyle{ \vartheta(0;q)^k = \vartheta_3^k(q) = \sum_{n=0}^{\infty}r_k(n)q^n, }[/math]

where

[math]\displaystyle{ \vartheta(0;q) = \sum_{n=-\infty}^{\infty}q^{n^2} = 1 + 2q + 2q^4 + 2q^9 + 2q^{16} + \cdots. }[/math]

Numerical values

The first 30 values for [math]\displaystyle{ r_k(n), \; k=1, \dots, 8 }[/math] are listed in the table below:

n	=	r1(n)	r2(n)	r3(n)	r4(n)	r5(n)	r6(n)	r7(n)	r8(n)
0	0	1	1	1	1	1	1	1	1
1	1	2	4	6	8	10	12	14	16
2	2	0	4	12	24	40	60	84	112
3	3	0	0	8	32	80	160	280	448
4	22	2	4	6	24	90	252	574	1136
5	5	0	8	24	48	112	312	840	2016
6	2×3	0	0	24	96	240	544	1288	3136
7	7	0	0	0	64	320	960	2368	5504
8	23	0	4	12	24	200	1020	3444	9328
9	32	2	4	30	104	250	876	3542	12112
10	2×5	0	8	24	144	560	1560	4424	14112
11	11	0	0	24	96	560	2400	7560	21312
12	22×3	0	0	8	96	400	2080	9240	31808
13	13	0	8	24	112	560	2040	8456	35168
14	2×7	0	0	48	192	800	3264	11088	38528
15	3×5	0	0	0	192	960	4160	16576	56448
16	24	2	4	6	24	730	4092	18494	74864
17	17	0	8	48	144	480	3480	17808	78624
18	2×32	0	4	36	312	1240	4380	19740	84784
19	19	0	0	24	160	1520	7200	27720	109760
20	22×5	0	8	24	144	752	6552	34440	143136
21	3×7	0	0	48	256	1120	4608	29456	154112
22	2×11	0	0	24	288	1840	8160	31304	149184
23	23	0	0	0	192	1600	10560	49728	194688
24	23×3	0	0	24	96	1200	8224	52808	261184
25	52	2	12	30	248	1210	7812	43414	252016
26	2×13	0	8	72	336	2000	10200	52248	246176
27	33	0	0	32	320	2240	13120	68320	327040
28	22×7	0	0	0	192	1600	12480	74048	390784
29	29	0	8	72	240	1680	10104	68376	390240
30	2×3×5	0	0	48	576	2720	14144	71120	395136

See also

• Jacobi's four-square theorem
• Gauss circle problem

External links

• Weisstein, Eric W.. "Sum of Squares Function". http://mathworld.wolfram.com/SumofSquaresFunction.html. 
• Sloane, N. J. A., ed. "Sequence A122141 (number of ways of writing n as a sum of d squares)". OEIS Foundation. https://oeis.org/A122141. 
• Sloane, N. J. A., ed. "Sequence A004018 (Theta series of square lattice, r_2(n))". OEIS Foundation. https://oeis.org/A004018.