Abstract
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.
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Chandrasekharan, K., Narasimhan, R.: The approximate functional equation for a class of zeta-functions. Math. Ann. 152, 30–64 (1963)
Elliott, P.D.T.A.: Central limit theorems for classical cusp forms. Ramanujan J. 36, 99–102 (2015)
Erdős, P., Kac, M.: On the Gaussian law of errors in the theory of additive functions. Proc. Nat. Acad. Sci. USA 25, 206–207 (1939)
Huxley, M.N.: On the difference between consecutive primes. Invent. Math. 15, 164–170 (1971)
E. Landau, Einführung in die Elementare umd Analytische Theorie der Algebraischen Zahlen und der Ideals, Chelsea Publishing Company (New York, 1949)
K. Liu and J. Wu, Weighted Erdős–Kac theorem in short intervals (preprint)
Lü, G.-S., Wang, Y.-H.: Note on the number of integral ideals in Galois extensions. Sci. China Math. 53, 2417–2424 (2010)
Lü, G.-S., Yang, Z.-S.: The average behavior of the coefficients of the Dedekind zeta function over square numbers. J. Number Theory 131, 1924–1938 (2011)
Nowak, W.G.: On the distribution of integer ideals in algebraic number fields. Math. Nachr. 161, 59–74 (1993)
G. Tenenbaum, Introduction to Analytic and Probabilistic Number Theory, Cambridge University Press (Cambridge, 1995)
Wu, J., Wu, Q.: Mean values for a class of arithmetic functions in short intervals. Math. Nachr. 293, 178–202 (2020)
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Liu, XL., Yang, ZS. Weighted Erdős–Kac Type Theorems Over Gaussian Field In Short Intervals. Acta Math. Hungar. 162, 465–482 (2020). https://doi.org/10.1007/s10474-020-01087-6
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DOI: https://doi.org/10.1007/s10474-020-01087-6