Primitive lattice points inside an ellipse

Werner Georg Nowak

Czechoslovak Mathematical Journal (2005)

  • Volume: 55, Issue: 2, page 519-530
  • ISSN: 0011-4642

Abstract

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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 ) ≤ 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 .

How to cite

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Nowak, Werner Georg. "Primitive lattice points inside an ellipse." Czechoslovak Mathematical Journal 55.2 (2005): 519-530. <http://eudml.org/doc/30966>.

@article{Nowak2005,
abstract = {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 $Q$.},
author = {Nowak, Werner Georg},
journal = {Czechoslovak Mathematical Journal},
keywords = {primitive lattice points; lattice point discrepancy; planar domains; primitive lattice points; lattice point discrepancy; planar domains},
language = {eng},
number = {2},
pages = {519-530},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Primitive lattice points inside an ellipse},
url = {http://eudml.org/doc/30966},
volume = {55},
year = {2005},
}

TY - JOUR
AU - Nowak, Werner Georg
TI - Primitive lattice points inside an ellipse
JO - Czechoslovak Mathematical Journal
PY - 2005
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 55
IS - 2
SP - 519
EP - 530
AB - 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 $Q$.
LA - eng
KW - primitive lattice points; lattice point discrepancy; planar domains; primitive lattice points; lattice point discrepancy; planar domains
UR - http://eudml.org/doc/30966
ER -

References

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