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2. Primitive lattice points inside an ellipse
- Creator:
- Nowak, Werner Georg
- Format:
- bez média and svazek
- Type:
- model:article and TEXT
- Subject:
- primitive lattice points, lattice point discrepancy, and planar domains
- Language:
- English
- Description:
- 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 $Q$.
- Rights:
- http://creativecommons.org/publicdomain/mark/1.0/ and policy:public