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 rk(n).

Definition

The function is defined as

\( {\displaystyle r_{k}(n)=|\{(a_{1},a_{2},\ldots ,a_{k})\in \mathbb {Z} ^{k}\ :\ n=a_{1}^{2}+a_{2}^{2}+\cdots +a_{k}^{2}\}|} \)

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

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

Formulae

k = 2
Main article: Sum of two squares theorem

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

\( {\displaystyle r_{2}(n)=4(d_{1}(n)-d_{3}(n))} \)

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:

\( {\displaystyle r_{2}(n)=4\sum _{d\mid n \atop d\,\equiv \,1,3{\pmod {4}}}(-1)^{(d-1)/2}} \)

The prime factorization \( {\displaystyle n=2^{g}p_{1}^{f_{1}}p_{2}^{f_{2}}\cdots q_{1}^{h_{1}}q_{2}^{h_{2}}\cdots } \) , where \( p_{i} \) are the prime factors of the form \) {\displaystyle p_{i}\equiv 1{\pmod {4}},} \) and \( q_{i} \) are the prime factors of the form \( {\displaystyle q_{i}\equiv 3{\pmod {4}}} \) gives another formula

\( {\displaystyle r_{2}(n)=4(f_{1}+1)(f_{2}+1)\cdots } \), if all exponents h 1 , h 2 , ⋯ {\displaystyle h_{1},h_{2},\cdots } {\displaystyle h_{1},h_{2},\cdots } are even. If one or more h i {\displaystyle h_{i}} h_{i} are odd, then r 2 ( n ) = 0 {\displaystyle r_{2}(n)=0} {\displaystyle r_{2}(n)=0}. \)

k = 3

Main article: Legendre's three-square theorem

Gauss proved that for a squarefree number n > 4,

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

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

k = 4
Main article: Jacobi's four-square theorem

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.

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

Representing n = 2km, where m is an odd integer, one can express \( {\displaystyle r_{4}(n)} \) in terms of the divisor function as follows:

\( {\displaystyle r_{4}(n)=8\sigma (2^{\min\{k,1\}}m).} \)

k = 8

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

\( {\displaystyle r_{8}(n)=16\sum _{d\,\mid \,n}(-1)^{n+d}d^{3}.} \)

Generating function

The generating function of the sequence \( {\displaystyle r_{k}(n)} \) for fixed k can be expressed in terms of the Jacobi theta function:[1]

\( {\displaystyle \vartheta (0;q)^{k}=\vartheta _{3}^{k}(q)=\sum _{n=0}^{\infty }r_{k}(n)q^{n},} \)

where

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

Numerical values

The first 30 values for \( {\displaystyle r_{k}(n),\;k=1,\dots ,8} \) 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