Alladi Ramakrishnan Hall

A sum worthy of Gauss

Chandan Dalawat

HRI, Allahabad

One looks at a certain sum $G$ involving the $p$-th roots of unity (where $p$ is
a prime number), called the quadratic Gaussian sum. It is easy to see that
$G^2=p$, which means that $G$ itself is either the positive or the negative
square root of $p$. Which one ?

It took Gauss many years to find the answer and to prove the result. Since
then some other proofs of this result have been given, and it has become the
central example of what is called the "root number" of an $L$-function. So
the result is very important.