Properties of Random Gauss Sums by Alicia Smith Reina ’22, Mathematics Senior Thesis Defense, Monday, May 16, 1:40 – 2:15 pm, North Science Building 113, Wachenheim.
Abstract: In his 1837 paper on primes in arithmetic progression, Dirichlet introduced a new function called the Dirichlet character. The real Dirichlet character of n mod p for p prime is known as the Legendre symbol and is defined to be 1 if n is a non-zero quadratic residue mod p, -1 if n is a quadratic non-residue mod p, and 0 if n is congruent to 0 mod p. When looking at the real Dirichlet character mod p for n between 1 and p-1 there is some noticeable structure, but it otherwise looks like a seemingly random string of 1s and -1s. However, this seemingly random string has the interesting property that the magnitude of its Gauss Sum is exactly the square root of p, where the Gauss Sum is a linear combination of the pth roots of unity with the coefficients given by the Dirichlet character. Inspired by this result, we study the properties and geometric behavior of the Gauss Sums of truly random strings of 1s and -1s, which I call Random Gauss Sums. In this thesis I will present some properties of the set of all Random Gauss Sums for a given prime p.