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 as the sum of 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.

Definition

The function is defined as
where denotes the cardinality of a set. In other words, is the number of ways can be written as a sum of 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.

Formulae

''k'' = 2

The number of ways to write a natural number as sum of two squares is given by. It is given explicitly by
where is the number of divisors of which are congruent to 1 modulo 4 and is the number of divisors of 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

''k'' = 3

Gauss proved that for a squarefree number,
where denotes the class number of an integer.
There exist extensions of Gauss' formula to arbitrary integer.

''k'' = 4

The number of ways to represent 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, where m is an odd integer, one can express in terms of the divisor function as follows:

''k'' = 6

The number of ways to represent as the sum of six squares is given by
where is the Kronecker symbol.

''k'' = 8

Jacobi also found an explicit formula for the case :

Generating function

The generating function of the sequence for fixed can be expressed in terms of the Jacobi theta function:
where

Numerical values

The first 30 values for are listed in the table below:
n=r1r2r3r4r5r6r7r8
0011111111
11246810121416
22041224406084112
330083280160280448
42224624902525741136
550824481123128402016
62×300249624054412883136
770006432096023685504
823041224200102034449328
9322430104250876354212112
102×508241445601560442414112
11110024965602400756021312
1222×3008964002080924031808
131308241125602040845635168
142×7004819280032641108838528
153×500019296041601657656448
16242462473040921849474864
1717084814448034801780878624
182×320436312124043801974084784
191900241601520720027720109760
2022×50824144752655234440143136
213×700482561120460829456154112
222×1100242881840816031304149184
232300019216001056049728194688
2423×30024961200822452808261184
2552212302481210781243414252016
262×13087233620001020052248246176
2733003232022401312068320327040
2822×700019216001248074048390784
2929087224016801010468376390240
302×3×5004857627201414471120395136