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# Forum Mathematicum

Managing Editor: Bruinier, Jan Hendrik

Ed. by Cohen, Frederick R. / Droste, Manfred / Darmon, Henri / Duzaar, Frank / Echterhoff, Siegfried / Gordina, Maria / Neeb, Karl-Hermann / Shahidi, Freydoon / Sogge, Christopher D. / Takayama, Shigeharu / Wienhard, Anna

IMPACT FACTOR 2018: 0.867

CiteScore 2018: 0.71

SCImago Journal Rank (SJR) 2018: 0.898
Source Normalized Impact per Paper (SNIP) 2018: 0.964

Mathematical Citation Quotient (MCQ) 2018: 0.71

Online
ISSN
1435-5337
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Volume 30, Issue 3

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

Wataru Takeda
/ Shin-ya Koyama
Published Online: 2017-10-06 | DOI: https://doi.org/10.1515/forum-2017-0152

## Abstract

We estimate the number of relatively r-prime lattice points in ${K}^{m}$ 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.

MSC 2010: 11N45; 11P21; 11R42; 52C07

## References

• [1]

S. J. Benkoski, The probability that k positive integers are relatively r-prime, J. Number Theory 8 (1976), no. 2, 218–223.

• [2]

J. Bourgain, Decoupling, exponential sums and the Riemann zeta function, J. Amer. Math. Soc. 30 (2017), no. 1, 205–224. 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.

• [4]

H. Iwaniec and E. Kowalski, Analytic Number Theory, Amer. Math. Soc. Colloq. Publ. 53, American Mathematical Society, Providence, 2004. Google Scholar

• [5]

S. Lang, Algebraic Number Theory, 2nd ed., Grad. Texts in Math. 110, Springer, New York, 1994. Google Scholar

• [6]

D. N. Lehmer, Asymptotic evaluation of certain totient sums, Amer. J. Math. 22 (1900), no. 4, 293–335.

• [7]

B. D. Sittinger, The probability that random algebraic integers are relatively r-prime, J. Number Theory 130 (2010), no. 1, 164–171.

• [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.

• [9]

W. Takeda, Visible lattice points and the extended Lindelöf hypothesis, J. Number Theory 180 (2017), 297–309.

Revised: 2017-09-15

Published Online: 2017-10-06

Published in Print: 2018-05-01

Citation Information: Forum Mathematicum, Volume 30, Issue 3, Pages 767–773, ISSN (Online) 1435-5337, ISSN (Print) 0933-7741,

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