Primitive lattice points in convex planar domains

Abstract

In this supposition, C ∞ can be replaced by C K with some K ∈ N sufficiently large. The whole somewhat technical condition is only stated to ensure the validity of the estimate (1.3), which in turn depends on the asymptotic expansion for the Fourier coefficients of the indicator function of √ x D (see Hlawka [6], [7]).

Visible Points on Curves over Finite Fields

“…These points have a natural geometric interpretation as points on F p,a (X, Y ) which are "visible" from the origin, see [2,6,7,10] and references therein for several other aspects of distribution of visible points in various regions. We show that on average over a = 0, .…”

Primitive lattice points in a thin strip along the boundary of a large convex planar domain

“…as was observed by Huxley & Nowak [7]. Improvements are only possible under the assumption of the Riemann Hypothesis (RH).…”

“…1. For the constant C we can give the alternative representation C = Z D (1/2) where Z D is the Hlawka zeta-function of the convex set D. (This is immediate, e.g., from the unnumbered formula below (3.5) in Huxley & Nowak [7].) Consequently, the main term can be written in the lucid form…”

“…), thereby refining the bound O(x 5/12+ε ) of [7]. If D is a circle, the error term can be improved to O(x 11/30+ε ), again under RH (see Zhai & Cao [15]).…”

Primitive lattice points in planar domains

“…Moroz [13], Hensley [7], Huxley and Nowak [10], Müller [14] and Zhai [21] have treated this question assuming the Riemann hypothesis (R.H.), with improving values of θ culminating in θ Z = 33349 84040 = 0.3968 . .…”