Czechoslovak Mathematical Journal, Vol. 72, No. 4, pp. 957-976, 2022
Weighted Erdős-Kac type theorem over quadratic field in short intervals
Xiaoli Liu, Zhishan Yang
Received June 1, 2021. Published online September 1, 2022.
Abstract: Let $\mathbb{K}$ be a quadratic field over the rational field and $a_{\mathbb{K}} ( n)$ be the number of nonzero integral ideals with norm $n$. We establish Erdős-Kac type theorems weighted by $a_{\mathbb{K}} (n)^l$ and $a_{\mathbb{K}} (n^2 )^l$ of quadratic field in short intervals with $l\in\mathbb{Z}^+$. We also get asymptotic formulae for the average behavior of $a_{\mathbb{K}}(n)^l$ and $a_{\mathbb{K}} ( n^2)^l$ in short intervals.
Keywords: ideal counting function; Erdős-Kac theorem; quadratic field; short intervals; mean value
