Sum of squares function

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.

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:

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

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

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