Startseite Estimates of lattice points in the discriminant aspect over abelian extension fields
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Estimates of lattice points in the discriminant aspect over abelian extension fields

  • Wataru Takeda ORCID logo EMAIL logo und Shin-ya Koyama
Veröffentlicht/Copyright: 6. Oktober 2017

Abstract

We estimate the number of relatively r-prime lattice points in Km with their components having a norm less than x, where K is a number field. The error terms are estimated in terms of x and the discriminant D of the field K, as both x and D grow. The proof uses the bounds of Dedekind zeta functions. We obtain uniform upper bounds as K runs through number fields of any degree under assuming the Lindelöf hypothesis. We also show unconditional results for abelian extensions with a degree less than or equal to 6.


Communicated by Jan Bruinier


References

[1] S. J. Benkoski, The probability that k positive integers are relatively r-prime, J. Number Theory 8 (1976), no. 2, 218–223. 10.1016/0022-314X(76)90103-7Suche in Google Scholar

[2] J. Bourgain, Decoupling, exponential sums and the Riemann zeta function, J. Amer. Math. Soc. 30 (2017), no. 1, 205–224. 10.1090/jams/860Suche in Google Scholar

[3] M. N. Huxley and N. Watt, Hybrid bounds for Dirichlet’s L-function, Math. Proc. Cambridge Philos. Soc. 129 (2000), no. 3, 385–415. 10.1017/S0305004100004527Suche in Google Scholar

[4] H. Iwaniec and E. Kowalski, Analytic Number Theory, Amer. Math. Soc. Colloq. Publ. 53, American Mathematical Society, Providence, 2004. 10.1090/coll/053Suche in Google Scholar

[5] S. Lang, Algebraic Number Theory, 2nd ed., Grad. Texts in Math. 110, Springer, New York, 1994. 10.1007/978-1-4612-0853-2Suche in Google Scholar

[6] D. N. Lehmer, Asymptotic evaluation of certain totient sums, Amer. J. Math. 22 (1900), no. 4, 293–335. 10.2307/2369728Suche in Google Scholar

[7] B. D. Sittinger, The probability that random algebraic integers are relatively r-prime, J. Number Theory 130 (2010), no. 1, 164–171. 10.1016/j.jnt.2009.06.008Suche in Google Scholar

[8] W. Takeda, The distribution of lattice points with relatively r-prime, preprint (2017), https://arxiv.org/abs/1704.02115, to appear in Algebra Discrete Math. Suche in Google Scholar

[9] W. Takeda, Visible lattice points and the extended Lindelöf hypothesis, J. Number Theory 180 (2017), 297–309. 10.1016/j.jnt.2017.04.002Suche in Google Scholar

Received: 2017-7-16
Revised: 2017-9-15
Published Online: 2017-10-6
Published in Print: 2018-5-1

© 2018 Walter de Gruyter GmbH, Berlin/Boston

Heruntergeladen am 8.9.2025 von https://www.degruyterbrill.com/document/doi/10.1515/forum-2017-0152/pdf
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