Brahmagupta's function

In number theory, Brahmagupta's function is a concept developed by the ancient Indian mathematician Brahmagupta around 628 CE.

Description

For positive integers $n$, Brahmagupta's function $h(n)$ is 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 $a,b,c,d\in \mathbb{\mathbb{Z}}$. The function holds particular importance in quadratic forms and has connections to modular forms.

Brahmagupta's function is multiplicative, 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 $p\equiv 1mod4$, and $h(p)=p-1$ when $p\equiv 3mod4$. This function played a crucial role in Jacobi's four-square theorem and influenced later work in arithmetic functions. The function has applications in cryptography and algebraic number theory.

References

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