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
Classification MSC:  11N60, 11N37, 11N45
DOI:  10.21136/CMJ.2022.0203-21


References:
[1] D. Bump: Automorphic Forms and Representations. Cambridge Studies in Advanced Mathematics 55. Cambridge University Press, Cambridge (1997). DOI 10.1017/CBO9780511609572 | MR 1431508 | Zbl 0868.11022
[2] K. Chandrasekharan, A. Good: On the number of integral ideals in Galois extensions. Monatsh. Math. 95 (1983), 99-109. DOI 10.1007/BF01323653 | MR 0697150 | Zbl 0498.12009
[3] K. Chandrasekharan, R. Narasimhan: The approximate functional equation for a class of zeta-functions. Math. Ann. 152 (1963), 30-64. DOI 10.1007/BF01343729 | MR 0153643 | Zbl 0116.27001
[4] P. Erdős, M. Kac: On the Gaussian law of errors in the theory of additive functions. Proc. Natl. Acad. Sci. USA 25 (1939), 206-207. DOI 10.1073/pnas.25.4.206 | Zbl 0021.20702
[5] M. N. Huxley: On the difference between consecutive primes. Invent. Math. 15 (1972), 164-170. DOI 10.1007/BF01418933 | MR 0292774 | Zbl 0241.10026
[6] E. Landau: Einführung in die elementare und analytische Theorie der algebraischen Zahlen und der Ideale. B. G. Teubner, Leipzig (1927). (In German.) MR 0031002 | JFM 53.0141.09
[7] X.-L. Liu, Z.-S. Yang: Weighted Erdős-Kac type theorems over Gaussian field in short intervals. Acta Math. Hung. 162 (2020), 465-482. DOI 10.1007/s10474-020-01087-6 | MR 4173309 | Zbl 1474.11169
[8] G. Lü, Y. Wang: Note on the number of integral ideals in Galois extensions. Sci. China, Math. 53 (2010), 2417-2424. DOI 10.1007/s11425-010-4091-7 | MR 2718837 | Zbl 1273.11160
[9] G. Lü, Z. Yang: The average behavior of the coefficients of Dedekind zeta function over square numbers. J. Number Theory 131 (2011), 1924-1938. DOI 10.1016/j.jnt.2011.01.018 | MR 2811559 | Zbl 1261.11073
[10] W. G. Nowak: On the distribution of integer ideals in algebraic number fields. Math. Nachr. 161 (1993), 59-74. DOI 10.1002/mana.19931610107 | MR 1251010 | Zbl 0803.11061
[11] J. Wu, Q. Wu: Mean values for a class of arithmetic functions in short intervals. Math. Nachr. 293 (2020), 178-202. DOI 10.1002/mana.201800276 | MR 4060372 | Zbl 07197944
[12] W. Zhai: Asymptotics for a class of arithmetic functions. Acta Arith. 170 (2015), 135-160. DOI 10.4064/aa170-2-3 | MR 3383642 | Zbl 1377.11106

Affiliations:   Xiaoli Liu, Zhishan Yang (corresponding author), Qingdao University, 308 Ningxia Rd., Qingdao 266071, Shandong, P. R. China, e-mail: 15216509811@163.com, zsyang@qdu.edu.cn


 
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