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).

The function is defined as

${\displaystyle r_{k}(n)=|\{(a_{1},a_{2},\ldots ,a_{k})\in \mathbb {Z} ^{k}\ :\ n=a_{1}^{2}+a_{2}^{2}+\cdots +a_{k}^{2}\}|}$

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

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

${\displaystyle r_{2}(n)=4(d_{1}(n)-d_{3}(n))}$

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:

${\displaystyle r_{2}(n)=4\sum _{d\mid n \atop d\,\equiv \,1,3{\pmod {4}}}(-1)^{(d-1)/2}}$

The prime factorization ${\displaystyle n=2^{g}p_{1}^{f_{1}}p_{2}^{f_{2}}\cdots q_{1}^{h_{1}}q_{2}^{h_{2}}\cdots }$, where ${\displaystyle p_{i}}$ are the prime factors of the form ${\displaystyle p_{i}\equiv 1{\pmod {4}},}$ and ${\displaystyle q_{i}}$ are the prime factors of the form ${\displaystyle q_{i}\equiv 3{\pmod {4}}}$ gives another formula

${\displaystyle r_{2}(n)=4(f_{1}+1)(f_{2}+1)\cdots }$, if all exponents ${\displaystyle h_{1},h_{2},\cdots }$ are even. If one or more ${\displaystyle h_{i}}$ are odd, then ${\displaystyle r_{2}(n)=0}$.

Gauss proved that for a squarefree number n > 4,

${\displaystyle r_{3}(n)={\begin{cases}24h(-n),&{\text{if }}n\equiv 3{\pmod {8}},\\0&{\text{if }}n\equiv 7{\pmod {8}},\\12h(-4n)&{\text{otherwise}},\end{cases}}}$

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

There exist extensions of Gauss' formula to arbitrary integer n.^[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.

${\displaystyle r_{4}(n)=8\sum _{d\,\mid \,n,\ 4\,\nmid \,d}d.}$

Representing n = 2^km, where m is an odd integer, one can express ${\displaystyle r_{4}(n)}$ in terms of the divisor function as follows:

${\displaystyle r_{4}(n)=8\sigma (2^{\min\{k,1\}}m).}$

The number of ways to represent n as the sum of six squares is given by

${\displaystyle r_{6}(n)=4\sum _{d\mid n}d^{2}{\big (}4\left({\tfrac {-4}{n/d}}\right)-\left({\tfrac {-4}{d}}\right){\big )},}$

where ${\displaystyle \left({\tfrac {\cdot }{\cdot }}\right)}$ is the Kronecker symbol.^[3]

Jacobi also found an explicit formula for the case k = 8:^[3]

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

The generating function of the sequence ${\displaystyle r_{k}(n)}$ for fixed k can be expressed in terms of the Jacobi theta function:^[4]

${\displaystyle \vartheta (0;q)^{k}=\vartheta _{3}^{k}(q)=\sum _{n=0}^{\infty }r_{k}(n)q^{n},}$

where

${\displaystyle \vartheta (0;q)=\sum _{n=-\infty }^{\infty }q^{n^{2}}=1+2q+2q^{4}+2q^{9}+2q^{16}+\cdots .}$

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

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

Grosswald, Emil (1985). Representations of integers as sums of squares. Springer-Verlag. ISBN 0387961267.

• Weisstein, Eric W. "Sum of Squares Function". MathWorld.
• Sloane, N. J. A. (ed.). "Sequence A122141 (number of ways of writing n as a sum of d squares)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
• Sloane, N. J. A. (ed.). "Sequence A004018 (Theta series of square lattice, r_2(n))". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.