Sum of squares function

Template:Short description 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. It is denoted by $r_k(n)$.

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, $r_k(n)$ is the number of ways $n$ can be written as a sum of $k$ 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 $r_2(n)$. It is given explicitly by

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

where $d_1(n)$ is the number of divisors of $n$ which are congruent to 1 modulo 4 and $d_3(n)$ is the number of divisors of $n$ 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 $n>4$,

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

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

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

k = 4

Template:Main 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.

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

Representing $n=2^{k}m$, 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 $n$ 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 $k=8$:

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

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	=	$r_1(n)$	$r_2(n)$	$r_3(n)$	$r_4(n)$	$r_5(n)$	$r_6(n)$	$r_7(n)$	$r_8(n)$
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