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Keywords:
ideal counting function; Erdős-Kac theorem; quadratic field; short intervals; mean value
ideal counting function; Erdős-Kac theorem; quadratic field; short intervals; mean value
Summary:
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.
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