Description
A Gaussian prime is a prime element in the ring of Gaussian integers $\mathbb{Z}[i]$. As the Gaussian integers lie on the plane, interesting questions about their geometric properties can be asked which have no classical analogue among the ordinary primes. Hecke proved that the Gaussian primes are equidistributed across sectors of the complex plane by making use of (infinite) Hecke characters and their associated $L$-functions. In this talk I will present several applications obtained upon applying the $L$-functions Ratios Conjecture to this family of $L$-functions. In particular, I will present a refined conjecture for the variance of Gaussian primes across sectors, and a conjecture for the one level density across this family. I will also discuss results related to “super even“ characters, which are the the function field analogue to Hecke characters.