Sum of squares function

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 square roots are counted as different, and is denoted by $r k (n)$ .

Definition

The function is defined as

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

where |.| denotes the cardinality of the 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 every sum has 2 sign combinations, and also $r_{2}(2)=4$ , since $2=(\pm 1)^{2}+(\pm 1)^{2}$ with 4 sign combinations. On the other hand is $r_{2}(3)=0$ , because there exists no way to represent 3 as a sum of two squares.

The first 20 values of r_k
n	r₁(n)	r₂(n)	r₃(n)	r₄(n)	r₅(n)	r₆(n)	r₇(n)	r₈(n)
0	1	1	1	1	1	1	1	1
1	2	4	6	8	10	12	14	16
2	0	4	12	24	40	60	84	112
3	0	0	8	32	80	160	280	448
4	2	4	6	24	90	252	574	1136
5	0	8	24	48	112	312	840	2016
6	0	0	24	96	240	544	1288	3136
7	0	0	0	64	320	960	2368	5504
8	0	4	12	24	200	1020	3444	9328
9	2	4	30	104	250	876	3542	12112
10	0	8	24	144	560	1560	4424	14112
11	0	0	24	96	560	2400	7560	21312
12	0	0	8	96	400	2080	9240	31808
13	0	8	24	112	560	2040	8456	35168
14	0	0	48	192	800	3264	11088	38528
15	0	0	0	192	960	4160	16576	56448
16	2	4	6	24	730	4092	18494	74864
17	0	8	48	144	480	3480	17808	78624
18	0	4	36	312	1240	4380	19740	84784
19	0	0	24	160	1520	7200	27720	109760

Particular cases

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 with 1 modulo 4 and $d 3 (n)$ is the number of divisors of $n$ which are congruent with 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 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

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

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

The generating series that gives the coefficients of the general form is based in terms of Jacobi theta function:^[1]

\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}}

References

^ Milne, Stephen C. (2002). "Introduction". Infinite Families of Exact Sums of Squares Formulas, Jacobi Elliptic Functions, Continued Fractions, and Schur Functions. Springer Science & Business Media. p. 9. ISBN 1402004915.

External links

Weisstein, Eric W. "Sum of Squares Function". MathWorld.

[1] Milne, Stephen C. (2002). "Introduction". Infinite Families of Exact Sums of Squares Formulas, Jacobi Elliptic Functions, Continued Fractions, and Schur Functions. Springer Science & Business Media. p. 9. ISBN 1402004915.

[1]

Definition

Particular cases

See also

References

External links