# 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. It is denoted by r$_k$(n).

## Definition

The function is defined as
    $r_{k}(n)=|\{(a_{1},a_{2},\ldots ,a_{k})\in \mathbb {Z} ^{k}\ :\ n=a_{1}^{2}+a_{2}^{2}+\cdots +a_{k}^{2}\}|$
where $|\,\ |$ 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

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:
    $r_{2}(n)=4\sum _{d\mid n \atop d\,\equiv \,1,3{\pmod {4}}}(-1)^{(d-1)/2}$
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 $p_{i}\equiv 1{\pmod {4}},$ and $q_{i}$ are the prime factors of the form $q_{i}\equiv 3{\pmod {4}}$ 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

Gauss proved that for a squarefree number n > 4,
    $r_{3}(n)={\begin{cases}24h(-n),&{\text{if }}n\equiv 3{\pmod {8}},\\0&{\text{if }}n\equiv 7{\pmod {8}},\\12h(-4n)&{\text{otherwise}},\end{cases}}$
where h(m) denotes the class number of an integer m.
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.
    $r_{4}(n)=8\sum _{d\,\mid \,n,\ 4\,\nmid \,d}d.$
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
    $r_{6}(n)=4\sum _{d\mid n}d^{2}{\big4\left({\tfrac {-4}{n/d}}\right)-\left({\tfrac {-4}{d}}\right){\big )},$
where $\left({\tfrac {\cdot }{\cdot }}\right)$ 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 k can be expressed in terms of the Jacobi theta function:
    $\vartheta (0;q)^{k}=\vartheta _{3}^{k}(q)=\sum _{n=0}^{\infty }r_{k}(n)q^{n},$
where
    $\vartheta (0;q)=\sum _{n=-\infty }^{\infty }q^{n^{2}}=1+2q+2q^{4}+2q^{9}+2q^{16}+\cdots .$

## Numerical values

The first 30 values for $r_{k}(n),\;k=1,\dots ,8$ 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 | 2$^2$ | 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 | 2$^3$ | 0 | 4 | 12 | 24 | 200 | 1020 | 3444 | 9328 |
| 9 | 3$^2$ | 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 | 2$^2$×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 | 2$^4$ | 2 | 4 | 6 | 24 | 730 | 4092 | 18494 | 74864 |
| 17 | 17 | 0 | 8 | 48 | 144 | 480 | 3480 | 17808 | 78624 |
| 18 | 2×3$^2$ | 0 | 4 | 36 | 312 | 1240 | 4380 | 19740 | 84784 |
| 19 | 19 | 0 | 0 | 24 | 160 | 1520 | 7200 | 27720 | 109760 |
| 20 | 2$^2$×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 | 2$^3$×3 | 0 | 0 | 24 | 96 | 1200 | 8224 | 52808 | 261184 |
| 25 | 5$^2$ | 2 | 12 | 30 | 248 | 1210 | 7812 | 43414 | 252016 |
| 26 | 2×13 | 0 | 8 | 72 | 336 | 2000 | 10200 | 52248 | 246176 |
| 27 | 3$^3$ | 0 | 0 | 32 | 320 | 2240 | 13120 | 68320 | 327040 |
| 28 | 2$^2$×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 |