Geodesic
Primitive lattice points inside an ellipse
Czechoslovak Mathematical Journal, Tome 55 (2005) no. 2, pp. 519-530
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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 $Q(u, v)\le x$, 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 $Q$.
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 $Q$.
Classification : 11E45, 11P21
Keywords: primitive lattice points; lattice point discrepancy; planar domains 
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Nowak, Werner Georg. Primitive lattice points inside an ellipse. Czechoslovak Mathematical Journal, Tome 55 (2005) no. 2, pp. 519-530. http://geodesic.mathdoc.fr/item/CMJ_2005_55_2_a22/

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