Draft:Brahmagupta's function

Brahmagupta's Function is a significant mathematical concept in number theory, developed by the ancient Indian mathematician Brahmagupta around 628 CE. The function is renowned as Brahmagupta's formula. This function, denoted as $h(n)$, is defined for positive integers $n$ and represents the number of proper representations of $n$ as a sum of four squares. Mathematically, it can be expressed as $h(n)=r_{4}(n)/8$, where $r_{4}(n)$ represents the total number of solutions to the equation $a^{2}+b^{2}+c^{2}+d^{2}=n$ with $a,b,c,d\in \mathbb {Z}$. The function holds particular importance in quadratic forms and has connections to modular forms.

A remarkable property of Brahmagupta's function is its multiplicative nature, meaning that for coprime numbers $m$ and $n$, we have $h(mn)=h(m)h(n)$. For prime numbers $p$, the function follows the formula $h(p)=p+1$ when $p\equiv 1{\pmod {4}}$, and $h(p)=p-1$ when $p\equiv 3{\pmod {4}}$. This function played a crucial role in Jacobi's four-square theorem and influenced later work in arithmetic functions. The function's significance extends to modern applications in cryptography and algebraic number theory.