Primitive lattice points inside an ellipse

Details Primitive lattice points inside an ellipse Primitive lattice points inside an ellipse

2003-07-21

arxiv.org/abs/math/0307279v1

Czechosl. Math. J. 55 (2005), 187-206

Mathematics

Number Theory

10 pages

Scientific paper

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 of R(x) is, in integral mean, at least a positive constant c times x^{1/4}. Furthermore, it is shown how to find an explicit value for c, for each specific given form Q.

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