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 Template:Math as the sum of Template:Math 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. It is denoted by Template:Math.

Definition

The function is defined as

$r_k(n)=|\{(a_1,a_2,\dots ,a_k)\in {\mathbb{Z}}^k:n=a_1^2+a_2^2+\cdots +a_k^2\}|$

where $||$ denotes the cardinality of a set. In other words, Template:Math is the number of ways Template:Math can be written as a sum of Template:Math squares.

For example, $r_2(1)=4$ since $1=0^2+(\pm 1{)}^2=(\pm 1{)}^2+0^2$ where each sum has two sign combinations, and also $r_2(2)=4$ since $2=(\pm 1{)}^2+(\pm 1{)}^2$ with four sign combinations. On the other hand, $r_2(3)=0$ 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

Template:Main The number of ways to write a natural number as sum of two squares is given by Template:Math. It is given explicitly by

$r_2(n)=4(d_1(n)-d_3(n))$

where Template:Math is the number of divisors of Template:Math which are congruent to 1 modulo 4 and Template:Math is the number of divisors of Template:Math which are congruent to 3 modulo 4. Using sums, the expression can be written as:

$r_2(n)=4\sum _{\genfrac{}{}{0ex}{}{d\mid n}{d\equiv 1,3(\mathrm{mod}4)}}(-1{)}^{(d-1)/2}$

The prime factorization $n=2^g{p}_1^{f_1}{p}_2^{f_2}\cdots q_1^{h_1}{q}_2^{h_2}\cdots$, where $p_i$ are the prime factors of the form $p_i\equiv 1(\mathrm{mod}4),$ and $q_i$ are the prime factors of the form $q_i\equiv 3(\mathrm{mod}4)$ gives another formula

$r_2(n)=4(f_1+1)(f_2+1)\cdots$, if all exponents $h_1,h_2,\cdots$ are even. If one or more $h_i$ are odd, then $r_2(n)=0$.

k = 3

Template:See also Gauss proved that for a squarefree number Template:Math,

$r_3(n)=\{\begin{array}{cc}24h(-n),& \text{if }n\equiv 3(\mathrm{mod}8),\\ 0& \text{if }n\equiv 7(\mathrm{mod}8),\\ 12h(-4n)& \text{otherwise},\end{array}$

where Template:Math denotes the class number of an integer Template:Math.

There exist extensions of Gauss' formula to arbitrary integer Template:Math.

k = 4

Template:Main The number of ways to represent Template:Math 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.

$r_4(n)=8\sum _{d\mid n,4\nmid d}d.$

Representing Template:Math, where m is an odd integer, one can express $r_4(n)$ in terms of the divisor function as follows:

$r_4(n)=8\sigma (2^{\min \{k,1\}}m).$

k = 6

The number of ways to represent Template:Math as the sum of six squares is given by

$r_6(n)=4\sum _{d\mid n}{d}^2(4\left(\frac{-4}{n/d}\right)-\left(\frac{-4}{d}\right)),$

where $\left(\frac{\cdot }{\cdot }\right)$ is the Kronecker symbol.

k = 8

Jacobi also found an explicit formula for the case Template:Math:

$r_8(n)=16\sum _{d\mid n}(-1{)}^{n+d}{d}^3.$

Generating function

The generating function of the sequence $r_k(n)$ for fixed Template:Math can be expressed in terms of the Jacobi theta function:

$\vartheta (0;q{)}^k={\vartheta }_3^k(q)=\sum _{n=0}^{\mathrm{\infty }}{r}_k(n)q^n,$

where

$\vartheta (0;q)=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{q}^{n^2}=1+2q+2q^4+2q^9+2q^{16}+\cdots .$

Numerical values

The first 30 values for $r_k(n),k=1,\dots ,8$ 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

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

External links

• Template:MathWorld
• Template:Cite OEIS
• Template:Cite OEIS