Distribution of lattice points visible from the origin

Question

I am reading Apostol's book on Introduction to Analytic Number Theory. In section 3.8, he discusses the distribution of lattice points visible from the origin. While finding the density of such points he considers squares. My question is: Would the answer be any different if we consider shapes other squares for instance circles or ellipses. I tried to find the answer for circles by inscribing and circumscribing squares but ended up with inequalities which hint that the answer might be the same. Any help would be greatly appreciated. Thanks.

Answer 1

From G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers (fourth edition 1960), pages 26, 29, 235, 237, 407, 409f. (lightly edited):

We call [$\ldots$] the system of points $(x, y)$ with integral coordinates the fundamental point-lattice $\Lambda.$ [$\ldots$] We call a point $P$ of $\Lambda$ visible (i.e. visible from the origin) if there is no point of $\Lambda$ on $OP$ between $O$ and $P.$ In order that $(x, y)$ should be visible, it is necessary and sufficient that $x/y$ should be in its lowest terms, or $(x, y) = 1.$

[$\ldots$]

Theorem 263. $$ \sum_{d\mid n}\mu(d) = 1 \quad (n = 1), \qquad \sum_{d\mid n}\mu(d) = 0 \quad (n > 1). $$

[$\ldots$]

Theorem 270. $$ g(x) = \sum_{m=1}^\infty f(mx) \iff f(x) = \sum_{n=1}^\infty\mu(n)g(nx). $$ The reader should have no difficulty in constructing a proof with the help of Theorem 263; but some care is required about convergence. A sufficient condition is that $$ \sum_{m,n}\left\lvert f(mnx)\right\rvert = \sum_kd(k)\left\lvert f(kx)\right\rvert $$ should be convergent. Here $d(k)$ is the number of divisors of $k.$ $\ \square$

[$\ldots$]

For every $\rho > 0,$ we denote by $\Lambda(\rho)$ the lattice of points $(\rho x, \rho y),$ where $x, y$ take all integral values, and write $g(\rho)$ for the number of points of $\Lambda(\rho)$ (apart from the origin $O$) which belong to the [given] bounded region $P.$ We call $$ V = \lim_{\rho\to0}\rho^2g(\rho) $$ the area of $P,$ if the limit exists. This definition embodies the only property of area which we require in what follows. It is clearly equivalent to any natural definition of area for elementary regions such as polygons, ellipses, etc.

[$\ldots$]

For our next result we require the idea of visible points of a lattice introduced in Ch. III. A point $T$ of $\Lambda(\rho)$ is visible (i.e. visible from the origin) if $T$ is not $O$ and if there is no point of $\Lambda(\rho)$ on $OT$ between $O$ and $T.$ We write $f(\rho)$ for the number of visible points of $\Lambda(\rho)$ belonging to $P$ and prove the following lemma.

Theorem 459. $$ \rho^2f(\rho) \to \frac{V}{\zeta(2)} \text{ as } \rho \to 0. $$

[$\ldots$]

The proof uses Theorem 270. It is omitted, partly because it takes about a page, partly because I'm worried about possibly infringing copyright, and partly because a generalisation to $\mathbb{R}^d$ for every integer $d \geqslant 2$ has been posted as the accepted answer to the MathOverflow question Reference request: probability that d numbers are coprime.