Geodesic
Weighted Erdős-Kac type theorem over quadratic field in short intervals
Czechoslovak Mathematical Journal, Tome 72 (2022) no. 4, pp. 957-976
Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

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.
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.
DOI : 10.21136/CMJ.2022.0203-21
Classification : 11N37, 11N45, 11N60
Keywords: ideal counting function; Erdős-Kac theorem; quadratic field; short intervals; mean value 
@article{10_21136_CMJ_2022_0203_21,
     author = {Liu, Xiaoli and Yang, Zhishan},
     title = {Weighted {Erd\H{o}s-Kac} type theorem over quadratic field in short intervals},
     journal = {Czechoslovak Mathematical Journal},
     pages = {957--976},
     year = {2022},
     volume = {72},
     number = {4},
     doi = {10.21136/CMJ.2022.0203-21},
     mrnumber = {4517587},
     zbl = {07655774},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.21136/CMJ.2022.0203-21/}
}
TY  - JOUR
AU  - Liu, Xiaoli
AU  - Yang, Zhishan
TI  - Weighted Erdős-Kac type theorem over quadratic field in short intervals
JO  - Czechoslovak Mathematical Journal
PY  - 2022
SP  - 957
EP  - 976
VL  - 72
IS  - 4
UR  - http://geodesic.mathdoc.fr/articles/10.21136/CMJ.2022.0203-21/
DO  - 10.21136/CMJ.2022.0203-21
LA  - en
ID  - 10_21136_CMJ_2022_0203_21
ER  - 
%0 Journal Article
%A Liu, Xiaoli
%A Yang, Zhishan
%T Weighted Erdős-Kac type theorem over quadratic field in short intervals
%J Czechoslovak Mathematical Journal
%D 2022
%P 957-976
%V 72
%N 4
%U http://geodesic.mathdoc.fr/articles/10.21136/CMJ.2022.0203-21/
%R 10.21136/CMJ.2022.0203-21
%G en
%F 10_21136_CMJ_2022_0203_21
Liu, Xiaoli; Yang, Zhishan. Weighted Erdős-Kac type theorem over quadratic field in short intervals. Czechoslovak Mathematical Journal, Tome 72 (2022) no. 4, pp. 957-976. doi: 10.21136/CMJ.2022.0203-21

[1] Bump, D.: Automorphic Forms and Representations. Cambridge Studies in Advanced Mathematics 55. Cambridge University Press, Cambridge (1997). | DOI | MR | JFM

[2] Chandrasekharan, K., Good, A.: On the number of integral ideals in Galois extensions. Monatsh. Math. 95 (1983), 99-109. | DOI | MR | JFM

[3] Chandrasekharan, K., Narasimhan, R.: The approximate functional equation for a class of zeta-functions. Math. Ann. 152 (1963), 30-64. | DOI | MR | JFM

[4] Erdős, P., Kac, M.: On the Gaussian law of errors in the theory of additive functions. Proc. Natl. Acad. Sci. USA 25 (1939), 206-207. | DOI | MR | JFM

[5] Huxley, M. N.: On the difference between consecutive primes. Invent. Math. 15 (1972), 164-170. | DOI | MR | JFM

[6] Landau, E.: Einführung in die elementare und analytische Theorie der algebraischen Zahlen und der Ideale. German B. G. Teubner, Leipzig (1927),\99999JFM99999 53.0141.09. | MR

[7] Liu, X.-L., Yang, Z.-S.: Weighted Erdős-Kac type theorems over Gaussian field in short intervals. Acta Math. Hung. 162 (2020), 465-482. | DOI | MR | JFM

[8] Lü, G., Wang, Y.: Note on the number of integral ideals in Galois extensions. Sci. China, Math. 53 (2010), 2417-2424. | DOI | MR | JFM

[9] Lü, G., Yang, Z.: The average behavior of the coefficients of Dedekind zeta function over square numbers. J. Number Theory 131 (2011), 1924-1938. | DOI | MR | JFM

[10] Nowak, W. G.: On the distribution of integer ideals in algebraic number fields. Math. Nachr. 161 (1993), 59-74. | DOI | MR | JFM

[11] Wu, J., Wu, Q.: Mean values for a class of arithmetic functions in short intervals. Math. Nachr. 293 (2020), 178-202. | DOI | MR | JFM

[12] Zhai, W.: Asymptotics for a class of arithmetic functions. Acta Arith. 170 (2015), 135-160. | DOI | MR | JFM

Cité par Sources :