Home Estimates of lattice points in the discriminant aspect over abelian extension fields
Article
Licensed
Unlicensed Requires Authentication

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

  • Wataru Takeda ORCID logo EMAIL logo and Shin-ya Koyama
Published/Copyright: October 6, 2017
Forum Mathematicum
From the journal Forum Mathematicum Volume 30 Issue 3

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.

Keywords: Lattice point; approximation formula; Lindelöf hypothesis in the discriminant aspect
MSC 2010: 11N45; 11P21; 11R42; 52C07

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)90103-7Search 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/860Search 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/S0305004100004527Search 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/053Search in Google Scholar

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

[6] D. N. Lehmer, Asymptotic evaluation of certain totient sums, Amer. J. Math. 22 (1900), no. 4, 293–335. 10.2307/2369728Search 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.2009.06.008Search 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. Search 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.2017.04.002Search 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

Articles in the same Issue

  1. Frontmatter
  2. A generalized uniqueness theorem and the graded ideal structure of Steinberg algebras
  3. Positive solutions for nonlinear nonhomogeneous parametric Robin problems
  4. Period relations for cusp forms of GSp4
  5. A generalization of a graph theory Mertens’ theorem: Galois covering case
  6. Commutators of the fractional integrals for second-order elliptic operators on Morrey spaces
  7. Vojta’s conjecture on rational surfaces and the abc conjecture
  8. The least unramified prime which does not split completely
  9. The ω-inequality problem for concatenation hierarchies of star-free languages
  10. Higher weight on GL(3). I: The Eisenstein series
  11. Fourier transforms of powers of well-behaved 2D real analytic functions
  12. Parabolic conformally symplectic structures I; definition and distinguished connections
  13. The quasi-arithmetic means and Cartan barycenters of compactly supported measures
  14. Estimates of lattice points in the discriminant aspect over abelian extension fields
  15. On Hecke eigenvalues of Siegel modular forms in the Maass space
  16. Harmonicity and minimality of complex and quaternionic radial foliations
https://doi.org/10.1515/forum-2017-0152
Keywords for this article
Lattice point; approximation formula; Lindelöf hypothesis in the discriminant aspect

Articles in the same Issue

  1. Frontmatter
  2. A generalized uniqueness theorem and the graded ideal structure of Steinberg algebras
  3. Positive solutions for nonlinear nonhomogeneous parametric Robin problems
  4. Period relations for cusp forms of GSp4
  5. A generalization of a graph theory Mertens’ theorem: Galois covering case
  6. Commutators of the fractional integrals for second-order elliptic operators on Morrey spaces
  7. Vojta’s conjecture on rational surfaces and the abc conjecture
  8. The least unramified prime which does not split completely
  9. The ω-inequality problem for concatenation hierarchies of star-free languages
  10. Higher weight on GL(3). I: The Eisenstein series
  11. Fourier transforms of powers of well-behaved 2D real analytic functions
  12. Parabolic conformally symplectic structures I; definition and distinguished connections
  13. The quasi-arithmetic means and Cartan barycenters of compactly supported measures
  14. Estimates of lattice points in the discriminant aspect over abelian extension fields
  15. On Hecke eigenvalues of Siegel modular forms in the Maass space
  16. Harmonicity and minimality of complex and quaternionic radial foliations
Sign up now to receive a 20% welcome discount
Institutional Access
How does access work?
Have an idea on how to improve our website?
Please write us.
© 2025 De Gruyter Brill
Downloaded on 10.9.2025 from https://www.degruyterbrill.com/document/doi/10.1515/forum-2017-0152/html
Scroll to top button