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Weighted Erdős-Kac type theorems over Gaussian field in short intervals. (English) Zbl 1474.11169

Summary: Assume that \(\mathbb{K}\) is Gaussian field, and \(a_{\mathbb{K}}(n)\) is the number of non-zero integral ideals in \(\mathbb{Z}[i]\) with norm \(n\). We establish an Erdős-Kac type theorem weighted by \(a_{\mathbb{K}}(n^2)^l\; (l\in\mathbb{Z}^+)\) in short intervals. We also establish an asymptotic formula for the average behavior of \(a_{\mathbb{K}}(n^2)^l\) in short intervals.

MSC:

11N60 Distribution functions associated with additive and positive multiplicative functions
11N37 Asymptotic results on arithmetic functions
11N45 Asymptotic results on counting functions for algebraic and topological structures
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