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).

DefinitionEdit

The function is defined as

 

where   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,   since   where each sum has two sign combinations, and also   since   with four sign combinations. On the other hand,   because there is no way to represent 3 as a sum of two squares.

FormulaeEdit

k = 2Edit

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

 

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:

 

The prime factorization  , where   are the prime factors of the form   and   are the prime factors of the form   gives another formula

 , if all exponents   are even. If one or more   are odd, then  .

k = 3Edit

Gauss proved that for a squarefree number n > 4,

 

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

k = 4Edit

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.

 

Representing n = 2km, where m is an odd integer, one can express   in terms of the divisor function as follows:

 

k = 8Edit

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

 

Generating functionEdit

The generating function of the sequence   for fixed k can be expressed in terms of the Jacobi theta function:[1]

 

where

 

Numerical valuesEdit

The first 30 values for   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

See alsoEdit

ReferencesEdit

  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.

External linksEdit