Theory article

The distribution of ideals whose norm divides $ n $ in the Gaussian ring

  • Received: 09 November 2023 Revised: 17 January 2024 Accepted: 22 January 2024 Published: 31 January 2024
  • MSC : 11M06, 11N99, 11R04

  • Let $ O_{K} = \mathbb{Z}[i] $. For each positive integer $ n $, denote $ \xi_{K}(n) $ as the number of integral ideals whose norm divides $ n $ in $ O_{K} $. In this paper, we studied the distribution of ideals whose norm divides $ n $ in $ O_{K} $ by using the Selberg-Delange method. This is a natural variant of a result studied by Deshouillers, Dress, and Tenenbaum (often called the DDT Theorem), and we found that the distribution function was subject to beta distribution with density $ \sqrt{3}/(2\pi\sqrt[3]{u^{2}(1-u)}) $.

    Citation: Tong Wei. The distribution of ideals whose norm divides $ n $ in the Gaussian ring[J]. AIMS Mathematics, 2024, 9(3): 5863-5876. doi: 10.3934/math.2024285

    Related Papers:

  • Let $ O_{K} = \mathbb{Z}[i] $. For each positive integer $ n $, denote $ \xi_{K}(n) $ as the number of integral ideals whose norm divides $ n $ in $ O_{K} $. In this paper, we studied the distribution of ideals whose norm divides $ n $ in $ O_{K} $ by using the Selberg-Delange method. This is a natural variant of a result studied by Deshouillers, Dress, and Tenenbaum (often called the DDT Theorem), and we found that the distribution function was subject to beta distribution with density $ \sqrt{3}/(2\pi\sqrt[3]{u^{2}(1-u)}) $.



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    [1] Z. Cui, J. Wu, The Selberg-Delange method in short intervals with an application, Acta Arith., 163 (2014), 247–260. http://dx.doi.org/10.4064/aa163-3-4 doi: 10.4064/aa163-3-4
    [2] H. Delange, Sur les formules dues Atle Selberg, Bull. Sci. Math., 83 (1959), 101–111.
    [3] H. Delange, Sur les formules de Atle Selberg, Acta Arith., 19 (1971), 105–146. https://doi.org/10.4064/AA-19-2-105-146 doi: 10.4064/AA-19-2-105-146
    [4] B. Feng, J. Wu, The arcsine law on divisors in arithmetic progressions modulo prime powers, Acta Math. Hungar., 163 (2021), 392–406. https://doi.org/10.1007/s10474-020-01105-7 doi: 10.1007/s10474-020-01105-7
    [5] B. Feng, J. Wu, $\beta$-law on divisors of integers representable as sum of two squares, in Chinese, Sci. China Math., 49 (2019), 1563–1572.
    [6] B. Feng, On the arcsine law on divisors in arithmetic progressions, Indagat. Math., 27 (2016), 749–763. https://doi.org/10.1016/j.indag.2016.01.008 doi: 10.1016/j.indag.2016.01.008
    [7] Keqin Feng, Algebraic Number Theory, in Chinese, Beijing: Science Press, 2000.
    [8] M. N. Huxley, The difference between consecutive primes, Invent. Math., 15 (1972), 164–170. https://doi.org/10.1007/BF01418933 doi: 10.1007/BF01418933
    [9] S. K. Leung, Dirichlet law for factorization of integers, polynomials and permutations, preprint paper, 2022.
    [10] A. Selberg, Note on the paper by L. G. Sathe, J. Indian Math. Soc., 18 (1954), 83–87. https://doi.org/10.18311/JIMS2F19542F17018 doi: 10.18311/JIMS2F19542F17018
    [11] G. Tenenbaum, Introduction to Analytic and Probabilistic Number Theory, 3 Eds, Cambridge: Cambridge University Press, 1995.
    [12] J. Wu, Q. Wu, Mean values for a class of arithmetic functions in short intervals, Math. Nachr., 293 (2020), 178–202. https://doi.org/10.1002/mana.201800276 doi: 10.1002/mana.201800276
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