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Title: Primitive lattice points inside an ellipse (English)
Author: Nowak, Werner Georg
Language: English
Journal: Czechoslovak Mathematical Journal
ISSN: 0011-4642 (print)
ISSN: 1572-9141 (online)
Volume: 55
Issue: 2
Year: 2005
Pages: 519-530
Summary lang: English
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Category: math
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Summary: Let $Q(u, v)$ be a positive definite binary quadratic form with arbitrary real coefficients. For large real $x$, one may ask for the number $B(x)$ of primitive lattice points (integer points $(m, n)$ with $\gcd (M,n) =1$) in the ellipse disc $Q(u, v)\le x$, in particular, for the remainder term $R(x)$ in the asymptotics for $B(x)$. While upper bounds for $R(x)$ depend on zero-free regions of the zeta-function, and thus, in most published results, on the Riemann Hypothesis, the present paper deals with a lower estimate. It is proved that the absolute value or $R(x)$ is, in integral mean, at least a positive constant $c$ time $x^{1/4}$. Furthermore, it is shown how to find an explicit value for $c$, for each specific given form $Q$. (English)
Keyword: primitive lattice points
Keyword: lattice point discrepancy
Keyword: planar domains
MSC: 11E45
MSC: 11P21
idZBL: Zbl 1081.11064
idMR: MR2137159
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Date available: 2009-09-24T11:25:16Z
Last updated: 2016-04-07
Stable URL: http://hdl.handle.net/10338.dmlcz/127999
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