Georgia Institute of Technology College of Sciences

We study the number $\Omega(n)$ of Gaussian prime factors of $n$, counted with multiplicity, when $n$ ranges over an angular sector of the Gaussian integers. We prove that, uniformly over the initial angle and over sector widths $\gamma\geq\gamma_N$, where $\gamma_N^{-1}=N^{o(1)}$, the distribution of $\Omega(n)$ is asymptotically shift-invariant. In concrete terms, the total variation between the proportions of Gaussian integers having $k$ and $k+1$ prime factors tends to zero, within a shrinking sector. If time permits, I'll mention some ergodic consequences. Joint work with Christina Giannitsi (Virginia Tech).