Weighted Erdős-Kac type theorem over quadratic field in short intervals

Abstract

Let $\mathbb{K}$ be a quadratic field over the rational field and ${a_\mathbb{K}}\left( n \right)$ be the number of nonzero integral ideals with norm n. We establish Erdős-Kac type theorems weighted by ${a_\mathbb{K}}{\left( n \right)^l}$ and ${a_\mathbb{K}}{\left( {{n^2}} \right)^l}$ of quadratic field in short intervals with l ∈ ℤ⁺. We also get asymptotic formulae for the average behavior of ${a_\mathbb{K}}{\left( n \right)^l}$ and ${a_\mathbb{K}}{\left( {{n^2}} \right)^l}$ in short intervals.