# 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

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 ${\displaystyle p_{i}}$  are the prime factors of the form ${\displaystyle p_{i}\equiv 1{\pmod {4}},}$  and ${\displaystyle q_{i}}$  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 ${\displaystyle h_{1},h_{2},\cdots }$  are even. If one or more ${\displaystyle h_{i}}$  are odd, then ${\displaystyle r_{2}(n)=0}$ .

### k = 3

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

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