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

:$r_k(n) = , \,$

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, $r_2(1) = 4$ 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 . It is given explicitly by

:$r_2(n) = 4(d_1(n)-d_3(n))$

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:

:$r_2(n) = 4\sum_(-1)^$
The prime factorization $n = 2^g p_1^p_2^\cdots q_1^q_2^\cdots$, 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 ,
:$r_3(n) = \begin 24 h(-n), & \text n\equiv 3\pmod, \\ 0 & \text n\equiv 7\pmod, \\ 12 h(-4n) & \text, \end$
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.
:$r_4(n)=8\sum_d.$

Representing , where ''m'' is an odd integer, one can express $r_4(n)$ in terms of the divisor function as follows:
:$r_4(n) = 8\sigma(2^m).$

''k'' = 6

The number of ways to represent as the sum of six squares is given by
:$r_6(n) = 4\sum_ d^2\big( 4\left(\tfrac\right) - \left(\tfrac\right)\big),$
where $\left(\tfrac\right)$ is the Kronecker symbol.

''k'' = 8

Jacobi also found an explicit formula for the case :
:$r_8(n) = 16\sum_(-1)^d^3.$

Generating function

The generating function of the sequence $r_k(n)$ for fixed can be expressed in terms of the Jacobi theta function:

:$\vartheta(0;q)^k = \vartheta_3^k(q) = \sum_^r_k(n)q^n,$

where

:$\vartheta(0;q) = \sum_^q^ = 1 + 2q + 2q^4 + 2q^9 + 2q^ + \cdots.$

Numerical values

The first 30 values for $r_k(n), \; k=1, \dots, 8$ are listed in the table below:

See also

* Integer partition
* Jacobi's four-square theorem
* Gauss circle problem

References

Further reading

External links

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*{{cite OEIS, A004018, Theta series of square lattice, r_2(n)

Arithmetic functions
Squares in number theory
Integer factorization algorithms