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Counting coprime pairs in random squares

José L. Fernández, Pablo Fernández

Abstract

Extending the classical Dirichlet's density theorem on coprime pairs, in this paper we describe completely the probability distribution of the number of coprime pairs in random squares of fixed side length in the lattice $\mathbb{N}^2$. The limit behaviour of this distribution as the side length of the random square tends to infinity is also considered.

Counting coprime pairs in random squares

Abstract

Extending the classical Dirichlet's density theorem on coprime pairs, in this paper we describe completely the probability distribution of the number of coprime pairs in random squares of fixed side length in the lattice . The limit behaviour of this distribution as the side length of the random square tends to infinity is also considered.
Paper Structure (10 sections, 4 theorems, 54 equations, 1 figure)

This paper contains 10 sections, 4 theorems, 54 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Notation and some background
  3. Some notations
  4. Probability and divisibility
  5. Inclusion/exclusion arguments
  6. Some results on arithmetic functions
  7. Inclusion/exclusion principle and Möbius function
  8. Cesàro's identity
  9. An auxiliary arithmetic function.
  10. The convolution Υ⋆μ$ We shall appeal to the convolution$Υ ⋆ μ$of the function$Υ $with the M\"{o}bius function$μ$. Notice that the function$Υ ⋆ μ$is multiplicative. For prime$p$, (\Upsilon \star \mu)(p)=\sum_{d \mid p} \mu(d)\, \Upsilon(p/d)= \mu(1) \,\Upsilon (p)+\mu(p)\,\Upsilon (1)=\Upsilon (p)-1=\frac{1}{p^2 - 2}, while for a prime power$p^k$with~$k 2$, (\Upsilon \star \mu)(p^k)= \sum_{d \mid p^k} \mu(d)\, \Upsilon(p^k/d) =\mu(1) \,\Upsilon (p^k)+\mu(p)\,\Upsilon (p^{k - 1})=\Upsilon (p) (\mu(1)+\mu(p))=0, using that$Υ$is strongly multiplicative. Thus, (\Upsilon \star\mu)(1)=1\quad \text{and}\quad (\Upsilon \star\mu)(n)=|\mu(n)| \prod_{p \mid n} \frac{1}{p^2 - 2} , \quad \hbox{for each $n \ge 2$}. Recall that$|μ(n)|=1$if$n$is square free, and is$0$otherwise. The convolution$(Υ ⋆μ)$is a non-negative function, and in fact we have the bounds |\mu(n)|\,\frac{1}{n^2}\le (\Upsilon \star\mu)(n)\le \frac{1}{\mathbf{F}}\, |\mu(n)|\,\frac{1}{n^2} , \quad \hbox{for any $n \ge 2$}. This follows from rewriting \ref{['eq:def de upsilon*mu']} as (\Upsilon \star\mu)(n)=\frac{|\mu(n)|}{n^2}\, \prod_{p \mid n} \frac{p^2}{p^2-2} , \quad \hbox{for any $n \ge 2$} , and from observing that 1< \prod_{p \mid n} \frac{p^2}{p^2-2}< \prod_{p} \frac{p^2}{p^2-2}=\frac{1}{\mathbf{F}} , \quad \hbox{for any $n \ge 2$}. Notice that both inequalities in \ref{['eq:bounds of OmegaStarMu']} are sharp: the constants 1 and$1/F$can not be improved; just take (big) prime numbers for the left inequality, and primorials for the right inequality. We consider now the average of$Υ ((i,j))$for$(i,j)∈ N_n^2$. Using Lemma \ref{['lemma:cesaro identity']}, we have that \sum_{1\le i,j \le n} \Upsilon (\gcd(i,j))=\sum_{k=1}^n (\Upsilon \star \mu)(k) \lfloor \frac{n}{k}\rfloor^2 , \quad \hbox{for any $n \ge 1$}. Writing$⌊ n/k⌋=n/k+{n/k}$, and using the bound \ref{['eq:bounds of OmegaStarMu']} and the values of$L_Υ (2)$from~\ref{['eq:value of Xi(2)']} and$L_μ(2)$from~\ref{['eq:1/zeta(2) con Mobius']}, we obtain that \begin{aligned} \sum_{1\le i,j \le n} \Upsilon (\gcd(i,j)) & =n^2 \sum_{k=1}^\infty \frac{(\Upsilon \star\mu)(k)}{k^2}+O(n) =n^2 \,L_{\Upsilon \star \mu}(2)+O(n) \\ & =n^2 \,L_{\Upsilon} (2) \,L_\mu(2)+O(n) =n^2 \,\frac{1}{\zeta(2)^2 \,\mathbf{F}}+O(n). \end{aligned} Thus \mathbf{E}_n(\Upsilon \circ \gcd)=\frac{1}{\zeta(2)^2 \,\mathbf{F}}+O(\frac{1}{n}) , \quad \hbox{for $n \ge 1$} , and, in particular, \lim_{n \to \infty} \mathbf{E}_n(\Upsilon \circ \gcd)=\frac{1}{\zeta(2)^2 \,\mathbf{F}}\, \cdot For$α, β ∈ (0,1]$, using that$Υ$is a positive function, we have that \sum_{\substack{1\le i \le \lfloor \alpha n\rfloor ,\\ 1\le j \le \lfloor \beta n\rfloor }} \Upsilon (\gcd(i,j)) \le \sum_{\substack{1\le i \le \alpha n,\\ 1\le j \le \beta n}} \Upsilon (\gcd(i,j)) \le \sum_{\substack{1\le i \le \lfloor \alpha n\rfloor +1,\\ 1\le j \le \lfloor \beta n\rfloor +1}} \Upsilon (\gcd(i,j)), and arguing as above, we deduce that \lim_{n\to \infty}\frac{1}{n^2}\sum_{\substack{1\le i \le \alpha n,\\ 1\le j \le \beta n}} \Upsilon (\gcd(i,j)) = (\alpha \beta) \,\frac{1}{\zeta(2)^2 \,\mathbf{F}} . More generally, for any$α, β,γ,δ ∈ (0,1]$such that$α>γ$and$β >δ$, \lim_{n\to \infty}\frac{1}{n^2}\sum_{\substack{\gamma n\le i \le \alpha n,\\ \delta n\le j \le \beta n}} \Upsilon (\gcd(i,j))= (\alpha-\gamma)( \beta-\delta) \,\frac{1}{\zeta(2)^2 \,\mathbf{F}} \cdot The next lemma shows how well distributed the values$Υ((i,j))$are. See FFequi1 for further connections between equidistribution and coprimality. If $f$ is a continuous function in the square $[0,1]^2$, then $\lim_{n \to \infty} \frac{1}{n^2} \sum_{\substack{0\le i/n\le 1, \\ 0\le j/n\le 1}} f(\frac{i}{n},\frac{j}{n})\, \Upsilon(\gcd(i,j)) = [\frac{1}{\zeta(2)^2\,\mathbf{F}}]\, \int_0^1 \!\!\int_0^1 f(x,y) \, dx \,dy.$ (The$i,j$in the sum above are integers.) This lemma claims that the sequence of probability measures$λ_n$in the square$[0,1]^2$given by \lambda_n:=\frac{1}{\Lambda_n}\sum_{\substack{0\le i/n\le 1, \\ 0\le j/n\le 1}} \delta_{({i}/{n},{j}/{n})}\Upsilon(\gcd(i,j)) , where$δ_(x,y)$denotes the point mass distribution at$(x,y)∈ [0,1]^2$, and where \Lambda_n=\sum_{\substack{0\le i/n\le 1, \\ 0\le j/n\le 1}} \Upsilon(\gcd(i,j)), converges weakly, as$n ∞$, to the Lebesgue measure in the square$[0,1]^2$. Fix an integer $k\ge 1$. The $i,j$ in the sums below are always integers. Denote, for integer $n > k$, $A_n:=\frac{1}{n^2} \sum_{\substack{0\le i/n\le 1, \\ 0\le j/n\le 1}} f(\frac{i}{n},\frac{j}{n})\,\Upsilon(\gcd(i,j)).$ For integers $u$ and $v$ such that $0\le u,v <k$, denote $A_n(u,v)=\frac{1}{n^2} \sum_{\substack{u/k\le i/n\le (u+1)/k, \\ v/k\le j/n\le (v+1)/k}} f(\frac{i}{n},\frac{j}{n})\,\Upsilon(\gcd(i,j)).$ Let $\phi(u,v)=\max\{f(x,y):\frac{u}{k}\le x\le \frac{u + 1}{k} \hbox{and} \frac{v}{k}\le y\le \frac{v + 1}{k}\} , \quad \hbox{for $0\le u,v <k$,}$ and $B_n(u,v)=\phi(u,v)\, \frac{1}{n^2}\sum_{\substack{u/k\le i/n\le (u+1)/k, \\ v/k\le j/n\le (v+1)/k}} \Upsilon(\gcd(i,j)).$ Because $\Upsilon$ is a positive function, we have that $A_n(u,v)\le B_n(u,v)$ for each $u$ and $v$. On account of \ref{['eq:equi de upsilon']}, we have that $\lim_{n \to \infty} B_n(u,v)=\phi(u,v) \, \frac{1}{k^2 \,\zeta(2)^2 \,\mathbf{F}}\cdot$ Since $A_n\le \sum_{\substack{0\le u <k,\\ 0\le v <k}}A_n(u,v)\le \sum_{\substack{0\le u <k,\\ 0\le v <k}}B_n(u,v) ,$ we have that $\limsup_{n\to \infty} A_n\le \frac{1}{\zeta(2)^2 \mathbf{F}} \, \frac{1}{k^2} \sum_{\substack{0\le u <k,\\ 0\le v <k}}\phi(u,v).$ This last inequality is valid for any integer $k \ge 1$, and so from $\lim_{k \to \infty}\frac{1}{k^2}\sum_{\substack{0\le u <k,\\ 0\le v <k}}\phi(u,v)= \int_0^1 \!\!\int_0^1 f(x,y) \,dx \, dy ,$ (by definition of the Riemann integral), we deduce that $\limsup_{n\to \infty} A_n\le \frac{1}{\zeta(2)^2\,\mathbf{F}} \int_0^1 \!\!\int_0^1 f(x,y) \,dx \,dy.$ Analogously, one obtains that $\liminf_{n\to \infty} A_n\ge \frac{1}{\zeta(2)^2\,\mathbf{F}} \int_0^1 \!\!\int_0^1 f(x,y) \,dx \,dy.\qedhere$ The following particular example of the lemma above, with$f(x,y)=(1-x)(1-y)$, will be used later in this paper: \lim_{n\to \infty} \frac{1}{n^2} \sum_{\substack{0\le i/n\le 1, \\ 0\le j/n\le 1}} (1-\frac{i}{n})( 1-\frac{j}{n})\,\Upsilon(\gcd(i,j)) =\frac{1}{4}\,\frac{1}{\zeta(2)^2\,\mathbf{F}} \cdot In this section, we study the distribution of the variable$Z_M$which counts coprime pairs in square windows of fixed side length$M$to obtain Theorem \ref{['teor:distribution of ZM']}, the main result of this paper. We denote by$V$the set of points$(n,m)$in$N^2$such that$(n,m)=1$, that is, so that$(n,m)$is a coprime pair. Points of$V$are frequently called \emph{visible points} (from the origin$(0,0)$), see, for instance, HerzogStewart. We denote the indicator function of$V$by$V$, so$V(n,m)=1$if$(n,m)=1$, and$V(n,m)=0$otherwise. We can write the function$V$as V(a,b)=\prod_{p } (1-I_p(a) \, I_p(b)) \quad\text{for $a,b\ge 1$}. Recall that$I_p(n)=1$if the prime$p$divides$n$, and$I_p(n)=0$otherwise. For each$(a,b)∈ N^2$, all but a finite number of factors in the above expression are equal to 1. In fact,$V(a,b)=1$if and only all factors are 1, or equivalently, if no prime$p$divides both$a$and$b$. In probabilistic terms, and using the notation of Section~\ref{['sec:probabilistic setting']}, equation \ref{['eq:lim-dirichlet']}, Dirichlet's density theorem, translates into \lim_{n \to \infty} \mathbf{E}_n(V)=\lim_{n\to \infty} \mathbf{P}_n(\mathcal{V})=\frac{1}{\zeta(2)}\cdot Fix a side length$M 1$, and denote by~$K_M$the square$K_M={1, …, M}^2$in$N^2$. For each point$(a,b)∈ N^2$, the window$W_M(a,b)$in the lattice$N^2$is the translation by~$(a,b)$of the square~$K_M$, that is, \mathcal{W}_M(a,b)=\{ a + 1, \ldots, a + M\}\times \{ b + 1, \ldots, b + M\}=(a,b) + \mathcal{K}_M. See again Figure \ref{['fig:window']}. We denote with$Z_M$the function defined in$N^2$which at each$(a,b)$gives the number of coprime pairs within the window$W_M(a,b)$, i.e., Z_M(a,b)=\sum_{(i,j)\in \mathcal{W}_M(a,b)} V(i,j). We prefer to write$Z_M$in the form Z_M(a,b)=\sum_{(k,l)\in \mathcal{K}_M} V(a + k,b + l) , so as to display the function$Z_M$as the sum of the$M^2$functions (a,b)\in \mathbb{N}^2 \mapsto V(a+k,b+l) , with$(k,l)$running over the square$K_M$. In principle, we have that$0 Z_M(a,b) M^2$, although the maximum value of$Z_M$is usually quite smaller than$M^2$; see Section~\ref{['section:pob distr of Z and Z*']}. The case $Z_M(a,b)=0$ would correspond to a $M$-window which is fully invisible (from the origin), in the sense that all points of the window have coordinates that are not coprime. There are explicit constructions of arbitrarily large 'invisible' squares; see, for instance, Theorem 5.29 in Apostol's book Apostol, or HerzogStewart. See also Section 5 in FFracsam for some related questions. The following explicit formula for$Z_M(a,b)$can be obtained by means of the inclusion/exclusion principle. Compare with Theorem 5 in~SugitaTakanobu. For $M\ge 1$ and $(a,b)\in\mathbb{N}^2$, $Z_M(a,b)=\sum_{d \ge 1} \mu(d)\lfloor \frac{M+r_d(a)}{d}\rfloor \lfloor \frac{M+r_d(b)}{d}\rfloor.$ Fix $M\ge 1$. For $d \ge 1$, let $C^{(d)}_M(a,b)=\{(i,j)\in (a,b)+\mathcal{K}_M: d\mid i\, \hbox{and} \, d \mid j\}.$ Now, $Z_M(a,b)=M^2-|\bigcup_{p\in \mathcal{P}} C^{(p)}_M(a,b)|.$ The union above is in fact a finite union, since for $p> a+M$, the set $C^{(p)}_M(a,b)$ is empty. Observe that, for prime $p$, $|C^{(p)}_M(a,b)|=\lfloor \frac{M+r_p(a)}{p}\rfloor \lfloor \frac{M+r_p(b)}{p}\rfloor ,$ and that, for primes $p$ and $q$, $|C^{(p)}_M(a,b)\cap C^{(q)}_M(a,b)|=|C^{(pq)}_M(a,b)|=\lfloor \frac{M+r_{pq}(a)}{pq}\rfloor \lfloor \frac{M+r_{pq}(b)}{pq}\rfloor ,$ and so on. Thus, \ref{['eq:formula ZM']} follows by \ref{['eq:inclusionexclusion mobius']} of Section \ref{['sec:inclusionexclusion mobius']}. Going back to the probabilistic setting in$N_n^2$, the random variable$Z_M$is a sum of$M^2$Bernoulli variables, see \ref{['eq:def of ZM']}. Each one of them has, asymptotically as$n ∞$, parameter$1/ζ(2)$, by Dirichlet's result \ref{['eq:lim-dirichlet-prob']}. All this readily gives, for the mean of$Z_M$, that \lim_{n \to \infty} \mathbf{E}_n(Z_M)=\frac{M^2}{\zeta(2)}\cdot But for the \emph{actual distribution} of$Z_M$, observe that those Bernoulli variables are not independent; in fact, they exhibit an interesting correlation structure, see Section \ref{['sec:correlation']}. \newline Our main interest here is to show that for each$M 1$, the random variable$Z_M$in$(N_n^2, P_n)$converges in distribution, as$n∞$, to a random variable$Z_M^⋆$taking values in${0,1,…,M^2}$, i.e., to prove that the limit \lim_{n \to \infty} \mathbf{P}_n(Z_M=r)\, =: \mathbf{P}(Z_M^\star=r) exists for each$r$such that$0 r M^2$and that$∑_r=0^M^2 P(Z_M^⋆=r)=1$. \newline The analysis in the following Sections~\ref{['sec:splitting']} and~\ref{['sec:distr of Z and Z*']} will provide, see Theorem~\ref{['teor:distribution of ZM']}, an \emph{explicit} and \emph{computable} formula for$P(Z_M^⋆=r)$for each$M 1$and all$0 r M^2$. There is a natural interaction between the side length~$M$of the square~$K_M$and divisibility properties of the points within the window$W_M(a,b)$. The (simple) reason is that a prime$p M$cannot divide simultaneously$a+k$and$a+k'$if$k,k'∈{1,…,M}$. So, it will be most convenient to separate, for each$(a,b)∈ N^2$, the points of$K_M$into two classes, as follows. Fix$M 1$and define P_M:=\prod_{p<M} p, with$P_1=P_2=1$. We shall assign, to each$(a,b)∈ N^2$, a pair of complementary subsets,$B_M(a,b)$and$A_M(a,b)$, within the square~$K_M$. We denote by$B_M(a,b)$the subset of$K_M$which consists of those pairs$(k,l)∈K_M$such that both$a + k$and$b + l$are divisible by some prime$p <M$: \mathcal{B}_M(a,b)= \bigcup_{p < M}\{(k,l) \in \mathcal{K}_M: a + k \equiv 0 \!\!\mod p \ \text{ and } \ b + l \equiv 0 \!\!\mod p\}. The set$A_M(a,b)$is just the complement of$B_M(a,b)$in$K_M$\mathcal{A}_M(a,b):=\mathcal{K}_M\setminus\mathcal{B}_M(a,b). Notice that$B_1(a,b)=B_2(a,b)=$, and also that, accordingly,$A_1(a,b)=K_1$and$A_2(a,b)=K_2$. It is a relevant fact that the sets$ B_M(a,b)$and$A_M(a,b)$depend only on the \emph{collection} of (pairs of) residues${(r_q(a), r_q(b)): q ∈ P_M}$, and thus, because of the Chinese remainder theorem, ultimately depend only on the pair of residues R_M(a,b):=(r_{P_M}(a),r_{P_M}(b)), Thus, there are~$P_M^2$different possibilities for the sets$A_M(a,b)$(and for the corresponding sets$B_M(a,b)$). We denote by$Φ_M(a,b)$the size of$A_M(a,b)$: \Phi_M(a,b)=|\mathcal{A}_M(a,b)|. Arguing with the inclusion/exclusion principle as in \ref{['eq:inclusionexclusion mobius']}, we obtain that \Phi_M(a,b)=\sum_{1 \le d \,\mid P_M} \mu(d) \lfloor \frac{M +r_d(a)}{d}\rfloor \lfloor\frac{M +r_d(b)}{d}\rfloor. The sum above extends to the divisors$d$of$P_M$. It is always the case, although far from sharp, that$Φ_M(a,b) M^2 -⌊(M - 1)/2⌋^2$. As mentioned before,$Φ_1(a,b)=1$and$Φ_2(a,b)=4$for all$(a,b)∈N^2$. For$M=3$, we have that$P_3=2$and that $\Phi_3(a,b)=8$ if both $a$ and $b$ are even, i.e., if $R_3(a,b)=(0,0)$;$\Phi_3(a,b)=5$ if both are odd, i.e., if $R_3(a,b)=(1,1)$;$\Phi_3(a,b)=7$ if one is even and the other is odd, i.e., if $R_3(a,b)$ is $(1,0)$ or $(0,1)$. We depict the four possible configurations of$A_3$and$B_3$in Figure \ref{['fig:cuadrados3']}. The four possible configurations of $\mathcal{A}_3$. The white squares represent the points of $\mathcal{A}_3(a,b)$; the blue (horizontally lined) squares, points of $\mathcal{B}_3(a,b)$, have both coordinates even. On top, we have noted $(a,b)$ modulo 2, and below each configuration, we have written the corresponding value of $\Phi_3(a,b)$, ranging from 5 to 8. For the case$M=4$, with$P_4=6$, Figure \ref{['fig:cuadrados4']} displays the 36 possible configurations. The 36 possible configurations of $\mathcal{A}_4$, labelled with the values of $(a,b)$ modulo 6, and the corresponding values of $\Phi_4(a,b)$, that in this case range from 9 to 12. As before, the white squares represent the points of $\mathcal{A}_4(a,b)$, and the blue (horizontally lined) squares correspond to points with both coordinates even; but now the red squares (vertically lined) have both coordinates divisible by 3. Some squares, of course, belong to both categories. For fixed$s$, such that$0 s M^2$, we denote with$ξ (M,s)$the (arithmetic) average of the binomial coeficientes$Φ_M(u,v) s$: \xi(M,s)=\frac{1}{P_M^2} \sum_{0\le u,v<P_M} \binom{\Phi_M(u,v)}{s} , \quad\hbox{for $0\le s \le M^2$}. Of course,$ξ(M,0)=1$. As we shall see later, in \ref{['eq:formula gamma(M,1)']}, the value of$ξ(M,1)$, which is the average value of$Φ_M(u,v)$, is \xi(M,1)=\frac{1}{P_M^2} \sum_{0\le u,v<P_M} \Phi_M(u,v)= M^2 \prod_{p<M} (1-\frac{1}{p^2}). Fix$M 1$. We use now the splitting of Section \ref{['sec:splitting']} to simplify the definition of$Z_M(a,b)$from the expression~\ref{['eq:def of ZM']} to Z_M(a,b)=\sum_{(k,l) \in \mathcal{K}_M} V(a + k,b + l) =\sum_{(k,l) \in \mathcal{B}_M(a,b)} V(a + k,b+l)+ \sum_{(k,l) \in \mathcal{A}_M(a,b)} V(a + k,b + l)=\sum_{(k,l) \in \mathcal{A}_M(a,b)} V(a + k,b + l) =\sum_{(k,l) \in \mathcal{A}_M(a,b)} \ \prod_{p} (1- I_p(a + k)\,I_p(b + l))=\sum_{(k,l) \in \mathcal{A}_M(a,b)} \ \prod_{p\ge M} (1- I_p(a + k)\,I_p(b + l)). Here, we have used \ref{['eq:formula V(a,b)']} and the very definitions of$A_M(a,b)$and$B_M(a,b)$. Notice also how this shows that the maximum possible value of$Z_M(a,b)$is$Φ(a,b)$, and not$M^2$. We now fix a pair$(u, v)$of residues modulo~$ P_M$, with$0 u,v < P_M$, Fix$n 1$. Recall, from \ref{['eq:def of RM']}, that$R_M(a,b)$denotes the pair$(r_P_M(a),r_P_M(b))$of residues of$(a,b)$modulo$P_M$. We are going to condition upon \Omega_n(u, v)=\{(a,b)\in \mathbb{N}_n^2: R_M(a,b)=(u, v)\}. There are a total of$P_M^2$different$Ω_n(u,v)$, which form a partition of$N_n^2$. Using Lemma \ref{['lemma:asympotic independence']}, we see that \lim_{n\to \infty} \mathbf{P}_n(\Omega_n(u,v))=\frac{1}{P_M^2} , \quad \hbox{for each $0 \le u,v <P_M$}\cdot For all$(a,b) ∈ Ω_n(u,v)$we have that$A_M(a,b)=A_M(u,v)$, since$R_M(a,b)=(u,v)$, that is,$a≡ u p$and$b≡ v p$for all prime$p>M$. Consequently, we have that$Φ_M(a,b)=Φ_M(u,v)$if$(a,b) ∈ Ω_n(u,v)$. \newline We apply now Lemma \ref{['lemma:schuette-nesbitt']}. Take, in the notation used there,$Ω_n(u,v)$as$Ω$, the function~$Z_M$as the counting function$C$and$Φ_M(u,v)$as$t$. Then, for any$(a,b)∈ Ω_n(u,v)$and for$r$such that$0 r Φ_M(u,v)$, we have that \begin{aligned} \mathop{\textup{\large1}}\nolimits_{\{Z_M=r\}}(a,b) & =\sum_{s=r}^{\Phi_M(u,v)} (-1)^{s-r} \binom{s}{r} \sum_{\substack{H \subset \mathcal{A}_M(u,v) \\ |H|=s}} \prod_{(k,l)\in H} \prod_{p\ge M} (1-I_p(a + k)\,I_p(b + l)). \\ & =\sum_{s=r}^{\Phi_M(u,v)} (-1)^{s-r} \binom{s}{r} \sum_{\substack{H \subset \mathcal{A}_M(u,v) \\ |H|=s}} \prod_{p\ge M} \prod_{(k,l)\in H} (1-I_p(a + k)\,I_p(b + l)). \end{aligned} Now observe that, if$(k,l)≠ (k', l') ∈ K_M$, then \begin{aligned} [1-I_p(a + k) & \,I_p(b+l)]\,[1-I_p(a + k')\,I_p(b + l')] \\ & \qquad\qquad =1-I_p(a + k)\,I_p(b + l)-I_p(a + k')\,I_p(b + l') , \end{aligned} because the term$I_p(a + k) I_p(b + l) I_p(a + k') I_p(b + l')$vanishes. This is so since if$k ≠ k'$, then$I_p(a + k) I_p(a + k')=0$, because the prime$p M$does not divide$k - k'$, and analogously, if$l≠ l'$, then$I_p(b + l) I_p(b + l')=0$. Using this observation, we may rewrite \ref{['eq:indicadora en Omegan1']} as follows: for any$(a,b)∈ Ω_n(u,v)$and for~$r$such that$0 r Φ(u,v)$, \mathop{\textup{\large1}}\nolimits_{\{ Z_M=r\}}(a,b) =\sum_{s=r}^{\Phi_M(u,v)} (-1)^{s-r}\! \binom{s}{r} \sum_{\substack{H \subset \mathcal{A}_M(u,v) \\ |H|=s}} \, \prod_{p\ge M}\!\! (1-\sum_{(k,l)\in H}\!\! I_p(a + k)\, I_p(b + l)) . Now, for each$p M$, the function$(a,b)↦ 1-∑_(k,l)∈ H I_p(a + k) I_p(b + l)$takes only the values 0 and 1. For if$p M$, then$p$may divide at most one$a+k$with$1 k M$(and also at most one$b+l$with$1 l M$). Thus this function is the indicator function of a certain subset in$N^2$, which we denote by$B_H^(p)$: \mathop{\textup{\large1}}\nolimits_{B_H^{(p)}}(a,b)=1-\sum_{(k,l)\in H} I_p(a + k)\, I_p(b + l). With this new notation, we may finally rewrite, for any$(a,b)∈ Ω_n(u,v)$and for$r$such that$0 r Φ(u,v)$, \mathop{\textup{\large1}}\nolimits_{\{Z_M=r\}}(a,b)=\sum_{s=r}^{\Phi_M(u,v)} (-1)^{s-r} \binom{s}{r} \sum_{\substack{H \subset \mathcal{A}_M(u,v) \\ |H|=s}} \, \prod_{p\ge M} \mathop{\textup{\large1}}\nolimits_{B_H^{(p)}}(a,b)=\sum_{s=r}^{\Phi_M(u,v)} (-1)^{s-r} \binom{s}{r} \sum_{\substack{H \subset \mathcal{A}_M(u,v) \\ |H|=s}} \, \mathop{\textup{\large1}}\nolimits_{\{\bigcap_{p\ge M} B_H^{(p)}\}}(a,b). Observe that for each$(a,b) ∈ N_n^2$, the above products (or the above intersections) are actually products/intersections of finitely many terms/sets, since for$p>n+M$, we have$I_p(a + k) I_p(b + l)=0$, for any$(k,l)∈ K_M$. Regarding these$B_H^(p)$, we have the following key lemma. For any $H \subset \mathcal{A}_M(u,v)$ such that $|H|=s$, we have that $\lim_{n \to \infty} \mathbf{P}_n(\bigcap_{p \ge M}B_H^{(p)}\,|\,\Omega_n(u,v))=\prod_{p\ge M} (1-\frac{s}{p^2}).$ This argument is modeled upon the proof of Theorem 5 in CaiBach. Fix $M\ge 1$ and a pair $(u,v)$ such that $0\le u,v<P_M$. Consider the sets $\Omega_n(u,v)$ and $\mathcal{A}_M(u,v)$ defined in \ref{['eq:def Omega_n']} and \ref{['eq:def A_m']}, respectively. Now fix a subset $H$ of $\mathcal{A}_M(u,v)$ of size $|H|=s$. Observe first that for $p \ge M$, and using the definition of $B_H^{(p)}$ in \ref{['eq:def de Bhp']}, \mathbf{P}_n(\mathbb{N}_n^2\setminus B_H^{(p)}\,|\,\Omega_n(u,v)) =\mathbf{E}_n (\sum_{(k,l)\in H} I_p(\cdot +k)\,I_p(\cdot+l) \,|\, \Omega_n(u,v))=\sum_{(k,l)\in H} \mathbf{E}_n(I_p(\cdot +k)\,I_p(\cdot+l) \,|\, \Omega_n(u,v))=\sum_{(k,l)\in H}\mathbf{P}_n(\{a,b\in\mathbb{N}^2_n: a+k\equiv 0 \textup{ mod }p, b+l\equiv 0 \textup{ mod }p\}\,|\, \Omega_n(u,v)). Using that $\Omega_n(u,v)$ is defined in terms of residues modulo the primes $q<M$, while the prime $p$ defining $B_H^{(p)}$ is $p\ge M$, Lemma \ref{['lemma:asympotic independence']} gives that $\lim_{n \to \infty}\mathbf{P}_n(\mathbb{N}^2_n\setminus B_H^{(p)}\,|\,\Omega_n(u,v))=\frac{s}{p^2}\cdot$ Analogously, for any distinct primes $p_1, \ldots, p_R$, all $\ge M$, we have that $\lim_{n \to \infty}\mathbf{P}_n(\bigcap_{i=1}^R (\mathbb{N}_n^2\setminus B_H^{(p_i)})\,|\,\Omega_n(u,v))=s^R\prod_{i=1}^R \frac{1}{p_i^2}\cdot$ Take $N>M$, and observe that $\mathbf{P}_n(\bigcap_{M\le p \le N}B_H^{(p)}\,|\,\Omega_n(u,v)) =\mathbf{P}_n(\mathbb{N}^2_n\setminus \bigcup_{M\le p \le N} (\mathbb{N}^2_n\setminus B_H^{(p)}) \,|\,\Omega_n(u,v)).$ From \ref{['eq:cota interseccion en primos']} and the inclusion/exclusion principle stated in \ref{['eq:IEprob']}, and as already discussed in Lemmas \ref{['lemma:cesaro identity']} and \ref{['lemma:formula ZM']}, we deduce that $\lim_{n \to \infty} \mathbf{P}_n(\bigcap_{M\le p \le N}B_H^{(p)}\,|\,\Omega_n(u,v))=\prod_{M\le p \le N} (1-\frac{s}{p^2}).$ As this holds for for any $N>M$, we deduce that $\limsup_{n \to \infty} \mathbf{P}_n(\bigcap_{ p \ge M}B_H^{(p)}\,|\,\Omega_n(u,v))\le\prod_{p\ge M} (1-\frac{s}{p^2}).$ For an inequality with $\liminf$ in the opposite direction, we argue as follows. For $k,l\le M$, we have that $\#\{(a,b)\in\mathbb{N}^2_n: a+k\equiv 0 \textup{ mod }p, b+l\equiv 0 \textup{ mod }p\} \le \#\{(c,d)\in\mathbb{N}^2_{n+M}: p\mid c \text{ and } p\mid d\},$ and so, $\mathbf{P}_n(\{(a,b)\in\mathbb{N}^2_n: a+k\equiv 0 \textup{ mod }p, b+l\equiv 0 \textup{ mod }p\}\,|\, \Omega_n(u,v)) \le \frac{1}{n^2}\, \lfloor \frac{n + M}{p}\rfloor^2 \frac{1}{1/P_M^2} \cdot$ This (rather crude) estimate is enough for our purposes. Now, going back to \ref{['eq:Pn de Nn menos BHp']}, we find that for some constant $C_M$, depending on $M$, and for $p \ge M$, $\mathbf{P}_n(\mathbb{N}_n^2\setminus B_H^{(p)}\,|\,\Omega_n(u,v))\le C_M \,\frac{s}{p^2}\cdot$ For $N >M$, we have that $\mathbf{P}_n(\bigcap_{M\le p \le N}B_H^{(p)}\,|\,\Omega_n(u,v))-\mathbf{P}_n(\bigcap_{ p \ge M}B_H^{(p)}\,|\,\Omega_n(u,v))=\mathbf{P}_n(\bigcup_{p \ge M}(\mathbb{N}^2_n\setminus B_H^{(p)})\,|\,\Omega_n(u,v)) -\mathbf{P}_n(\bigcup_{M\le p \le N}(\mathbb{N}^2_n\setminus B_H^{(p)})\,|\,\Omega_n(u,v))=\mathbf{P}_n(\bigcup_{p >N}(\mathbb{N}^2_n\setminus B_H^{(p)})\,|\,\Omega_n(u,v)) \le \sum_{p >N} \mathbf{P}_n((\mathbb{N}_n^2\setminus B_H^{(p)})\,|\,\Omega_n(u,v)) \le C_M \, s\, \sum_{p>N}\frac{1}{p^2} ,$ where \ref{['eq:cota Pn de Nn menos BHp']} was used in the last inequality. Thus, we can deduce, using \ref{['eq:interseccion BHp condic']}, that, for $N >M$, $\liminf_{n \to \infty} \mathbf{P}_n(\bigcap_{ p \ge M}B_H^{(p)}\,|\,\Omega_n(u,v))\ge\prod_{M\le p \le N} (1-\frac{s}{p^2})-C_M \, s\, \sum_{p>N}\frac{1}{p^2} ,$ and conclude, upon letting $N \to \infty$, that $\liminf_{n \to \infty} \mathbf{P}_n(\bigcap_{ p \ge M}B_H^{(p)}\,|\,\Omega_n(u,v))\ge \prod_{p\ge M} (1-\frac{s}{p^2}) ,$ which finishes the proof. We derive now the probability distribution of$Z_M^⋆$by means of Lemma \ref{['lemma:a la caibach']}. From Lemma \ref{['lemma:a la caibach']} and the expression \ref{['eq:indicadora en Omegan3']}, we deduce, for each$(u,v)$such that$0 u,v <P_M$, that \begin{aligned} \lim_{n \to \infty} \mathbf{P}_n(Z_M=r\, |\,\Omega_n(u,v)) & =\sum_{s=r}^{\Phi_M(u,v)} (-1)^{s-r} \binom{s}{r} \sum_{\substack{H \subset \mathcal{A}_M(u,v) \\ |H|=s}} \prod_{p \ge M}(1-\frac{s}{p^2}) \\ & =\sum_{s=r}^{\Phi_M(u,v)} (-1)^{s-r} \binom{s}{r} \binom{\Phi(u,v)}{s} \prod_{p \ge M} (1-\frac{s}{p^2}). & \end{aligned} Therefore, from total probability and \ref{['eq:lim prob Omega_n']}, we finally conclude that \begin{aligned} \mathbf{P}(Z^{\star}_M=r) & := \lim_{n \to \infty} \mathbf{P}_n(Z_M=r) \\ & = \frac{1}{P_M^2} \sum_{0\le u,v<P_M} \sum_{s=r}^{\Phi_M(u,v)} (-1)^{s-r} \binom{s}{r}\binom{\Phi_M(u,v)}{s} \prod_{p \ge M} (1-\frac{s}{p^2}) \\ & = \frac{1}{P_M^2} \sum_{0\le u,v<P_M} \sum_{s=r}^{M^2} (-1)^{s-r} \binom{s}{r}\binom{\Phi_M(u,v)}{s} \prod_{p \ge M} (1-\frac{s}{p^2}). \end{aligned} Part of the conclusion is that each of the limits$_n ∞ P_n(Z_M=r)$exists; the other, of course, is the precise values of those limits. The values of these limits define the probability distribution of a variable~$Z_M^⋆$with values in${0,…, M^2}$. The sum of the above probabilities as$r$runs from$r=0$to$r=M^2$is 1, as it should, simply because$∑_r=0^s (-1)^s - r sr=0$, unless$s=0$, in which case it is 1. With the notations above, including that recorded in \ref{['eq:formula of gammas']}, we have for $M \ge 1$ and $0 \le r \le M^2$ that $\mathbf{P}(Z^{\star}_M=r)=\sum_{s=r}^{M^2} (-1)^{s-r} \binom{s}{r}\,\xi(M,s) \prod_{p \ge M} (1-\frac{s}{p^2}).$ Thus, as$n ∞$, the variable$Z_M$in the probability space$(N^2_n, P_n)$converges in distribution to the random variable$Z_M^⋆$with probability mass function given by \ref{['eq:prob distr de ZMstar']}. As we have already pointed out, the case$M = 1$is Dirichlet's density theorem: the variable$Z^⋆_1$is a Bernoulli variable with success parameter$P(Z_1^⋆=1)=∏_p (1-s/p^2)=1/ζ(2)$. \newline The formula in \ref{['eq:prob distr de ZMstar']} give the following probabilities, rounded to two decimal places, for the case$M=2$: \text{\footnotesize$\begin{array}{c|cccccccccccccccccccc} r & 0 & 1 & 2 & 3 & 4 \\ \mathbf{P}(Z^{\star}_2=r) & 0.21\% & 6.59\% & 43.00\% & 50.20\% & - \end{array}$} The values for the case$M=3$are: \text{\footnotesize$\begin{array}{c|cccccccccccccccccccc} r & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & \\ \mathbf{P}(Z^{\star}_3=r) & 0.00\% & 0.02\% & 0.48\% & 4.74\% & 16.21\% & 24.41\% & 35.20\% & 17.71\% & 1.23\% & - \end{array}$} And for the case$M=4$, \text{\footnotesize$\begin{array}{c|cccccccccccccccccccc} r & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & \\ \mathbf{P}(Z^{\star}_4=r) & 0.00\% & 0.00\% & 0.00\% & 0.00\% & 0.00\% & 0.01\%\% & 0.27\% & 2.37\% & 10.67\% \\ r & 9 & 10 & 11 & 12 & 13 & 14 & 15 & 16 \\ \mathbf{P}(Z^{\star}_4=r) & 25.83\% & 35.68\% & 22.18\% & 2.99\% & - & - & - & - \end{array}$} In all these tables, a dash "$-$" means probability (exactly) 0: not attainable values. These are the graphical representations of the mass functions for$M=3,4,5$in a common range${0,1,…,25}$: Mass function of $Z^*_3$. Mass function of $Z^*_4$. Mass function of $Z^*_5$. For the mean of$Z_M^⋆$we have, using \ref{['eq:lim mean of ZM in Nn']}, that \begin{aligned} \mathbf{E}(Z_M^\star) & =\sum_{r=0}^{M^2} r \,\mathbf{P}(Z_M^\star=r) =\sum_{r=0}^{M^2} r \lim_{n \to \infty}\mathbf{P}_n(Z_M=r) \\ & =\lim_{n \to \infty}\sum_{r=0}^{M^2} r\,\mathbf{P}_n(Z_M=r)=\lim_{n \to \infty} \mathbf{E}_n(Z_M)=\frac{M^2}{\zeta(2)} \cdot \end{aligned} For the probability generating function$G_Z_m^⋆$of$Z^⋆_M$, we obtain immediately from \ref{['eq:prob distr de ZMstar']} and the binomial theorem the following. See also Lemma~\ref{['lemma:inclusion/exclusion and pgf']}. For $|z|\le 1$, $G_{Z_m^\star}(z):= \sum_{r=0}^{M^2} \mathbf{P}(Z^\star_M=r) \,z^r=\sum_{s=0}^{M^2} (z-1)^s\, \xi(M,s) \,\prod_{p\ge M}(1-\frac{s}{p^2}).$ For$z=1$, both sides of the expression \ref{['eq:pgf of ZMstar']} of the probability generating function of~$Z_M^⋆$give 1. Differentiating$G_Z_m^⋆(z)$and evaluating at$z=1$, we get \mathbf{E}(Z^\star_M)=\xi(M,1) \prod_{p\ge M}(1-\frac{1}{p^2}), and so, \ref{['eq:mean of ZMstar']} yields \xi(M,1)= M^2 \prod_{p <M} (1-\frac{1}{p^2}) . In general, by repeated differentiation of$G_Z_m^⋆(z)$, we get, for the factorial moments, \mathbf{E}(\binom{Z_M^\star}{s})= \mathbf{E}(\frac{1}{s!}\,(Z_M^\star(Z_M^\star-1)\cdots (Z_M^\star-s+1))=\xi(M,s) \prod_{p \ge M} (1-\frac{s}{p^2})=\frac{M^2}{\zeta(2)}\,\frac{\xi(M,s)}{\xi(M,1)}\quad\text{for $0 \le s \le M^2$.} For each $M$, it appears that $\Phi_M(a,b)$ takes few different values concentrated around its mean value $\xi(M,1)$ given by \ref{['eq:formula gamma(M,1)']}. For example, we have that $\xi(3,1)=6\hbox{.}8$, while the values of $\Phi_3(a,b)$ range from 5 to 8; for $M=4$, $\xi(4,1)=10\hbox{.}7$, and $\Phi_4(a,b)$ takes values between 9 and 12. See Figures \ref{['fig:cuadrados3']} and \ref{['fig:cuadrados4']}. Assume for the sake of the argument that $\Phi_M(a,b)$ were to be constantly the integer $\xi(M,1)$. If this were the case, then we would have, from \ref{['eq:formula of gammas']}, $\xi(M,s)=\binom{\xi(M,1)}{s} ,\quad \hbox{for each $s\ge 0$} ,$ and thus, plugging this into \ref{['eq:prob distr de ZMstar']}, the probability $\mathbf{P}(Z_M^\star=r)$ would be, approximately, $\sum_{s=r}^{\xi(M,1)} (-1)^{s - r} \binom{s}{r} \binom{\xi(M,1)}{s} \prod_{p\ge M} (1-\frac{s}{p^2}) , \quad \hbox{for $0\le r \le M^2$}.$ Further, approximate $\prod_{p\ge M} (1-\frac{s}{p^2}) \approx \prod_{p\ge M} (1-\frac{1}{p^2})^s , \quad \hbox{for each $s \ge 0$} ,$ and denote $q_M:=\prod_{p\ge M} (1-\frac{1}{p^2}).$ With all this, for $0\le r \le M^2$, $\mathbf{P}(Z_M^\star=r)$ would be, approximately, $\sum_{s=r}^{\xi(M,1)} (-1)^{s - r} \binom{s}{r} \binom{\xi(M,1)}{s} \,q_M^s,$ and then $Z_m^\star$ would follow a binomial distribution with $\xi(M,1)$ repetitions and success probability $q_M$. Or better, since $\xi(M,1) q_M={M^2}/{\zeta(2)}$ (see \ref{['eq:formula gamma(M,1)']}), we could expect the variable $Z_M^\star$ to be relatively close to a Poisson distribution with parameter $M^2/\zeta(2)$, with closeness depending upon the factorization of $M$. This section is devoted to study the correlation structure between the random variables that register coprimality of pairs of points in$N^2$. This will be used later to obtain estimates of the variance of$Z_M^⋆$. For $(a,b) \in \mathbb{N}^2$ and integers $i,j \ge 0$, we have that $V(a,b)\,V(a+i,b+j)=\prod_{p\,\mid \,\gcd(i,j)} (1-I_p(a)\,I_p(b)) \prod_{p \,\nmid \,\gcd(i,j)}(1-I_p(a)\,I_p(b)-I_p(a+i)\,I_p(b+j)).$ It follows from writing $V(a,b)\, V(a+i,b+j) =\prod_{p} (1-I_p(a)\,I_p(b)) \,(1-I_p(a+i)\,I_p(b+j))\qquad=\prod_{p} (1-I_p(a)\,I_p(b)-I_p(a+i)\,I_p(b+j)+I_p(a)\,I_p(b) \,I_p(a+i)\,I_p(b+j)) ,$ and from observing that if $p \mid i$ then $I_p(a)=I_p(a+i)$, and that if $p \nmid i$ then $I_p(a)\,I_p(a+i)=0$, and analogously for $b$ and $j$. For$i,j 0$fixed, consider$(a,b) ∈ N_n^2 ↦ V(a,b)V(a+i,b+j)$; it follows from Lemma~\ref{['lemma:correlation']}, that \lim_{n \to \infty} \mathbf{E}_n(V(\cdot, \cdot) \, V(\cdot+i,\cdot+j))=\prod_{p \,\mid \,\gcd(i,j)} (1-\frac{1}{p^2})\, \prod_{p \,\nmid \,\gcd(i,j)} (1-\frac{2}{p^2})=\mathbf{F}\, \Upsilon (\gcd(i,j)). In general, and analogously, we have the following. For $i,j \ge 0$ and $k,l \ge 0$, we have that $\lim_{n \to \infty} \mathbf{E}_n(V(\cdot+k, \cdot+l) \,V(\cdot+i,\cdot+j))=\mathbf{F} \, \Upsilon (\gcd(|i-k|,|j-l|)).$ Consequently, for the coefficient of correlation, we find that \lim_{n \to \infty}\rho_n (V(\cdot+k, \cdot+l), V(\cdot+i,\cdot+j))\qquad =\frac{\zeta(2)^2\, \mathbf{F}\, \Upsilon (\gcd(|i-k|,|j-l|)) -1 }{\zeta(2)-1}=:\rho(i,j,k,l). Recall that$(0,0)=0$and that$Υ(0)=1/(F ζ(2)$, and observe that \ref{['eq:precovariance']} gives that \lim_{n \to \infty} \mathbf{E}_n(V(\cdot+k, \cdot+l) V(\cdot+k,\cdot+l))=\mathbf{F} \Upsilon (0)=\frac{1}{\zeta(2)} , as we already know, see \ref{['eq:dirichlet general kl']}. On account of the bounds for the function$Υ$in \ref{['eq:cotas de Upsilon']}, we see that all the (limit) coefficients of correlation$ρ(i,j,k,l)$defined in \ref{['eq:formula correlacion']} satisfy \frac{\zeta(2)^2\,\mathbf{F}-1}{\zeta(2)-1} \le \rho(i,j,k,l)\le 1. This (somewhat mysterious) lower bound, with numerical value$≈ -0.19694$, is attained when$Υ(1)=1$is plugged into \ref{['eq:formula correlacion']}, that is, when$(|i-k|,|j-l|)=1$; and this happens with probability$1/ζ(2)≈ 60.79%$. The second most likely value is$≈ 0.4799$, and occurs when~$Υ$takes the value$3/2$. This happens (see \ref{['es:def de Upsilon']}, and also the list of values in~\ref{['es:values of Upsilon']}) when$(|i-k|,|j-l|)$is a power of 2. As the probability that$(a,b)=k$, for integers~$a$and~$b$, is$1/(k^2ζ(2))$, a quick calculation gives that the value$≈ 0.4799$is taken with probability$1/(3ζ(2))≈ 20.26%$. In fact, as we verify next, the (asymptotic) \emph{average correlation} is 0. $\lim_{N\to\infty} \frac{1}{N^4} \sum_{\substack{(k,i), (l,j)\in \mathcal{K}_N}}\rho(i,j,k,l) = 0.$ For integer $N$, consider $A_N:=\sum_{\substack{(k,i), (l,j)\in \mathcal{K}_N}} \Upsilon(\gcd(|i-k|, |j-l|)).$ Classify now $(k,i) \in \mathcal{K}_{N}$ according to whether $k<i$, $k=i$, or $k >i$, and the same with $(l,j)$, to obtain that $A_N=\frac{N^2}{\zeta(2)} +4 \,(N\sum_{c=1}^{N} (N-c)\,\Upsilon(c) +\sum_{1 \le c,d \le N} (N-c)(N-d)\, \Upsilon (\gcd(c,d))).$ Recall, \ref{['eq:cotas de Upsilon']}, that the function $\Upsilon$ satisfies $1\le \Upsilon (n)\le \frac{1}{\zeta(2) \mathbf{F}}$, for any $n \ge 1$. Therefore, we have that $N \sum_{c=1}^{N} (N-c)\,\Upsilon(c)\le \frac{1}{\zeta(2)\, \mathbf{F}} \,N^3 ,$ and thus, $A_N= 4 \sum_{1 \le c,d \le N} (N-c)(N-d) \,\Upsilon (\gcd(c,d))+O(N^3).$ As shown in \ref{['eq:promedio (1-x)(1-y) upsilon']}, we have that $\frac{1}{N^2} \sum_{1 \le c,d \le N} (1-\frac{c}{N})(1-\frac{d}{N}) \,\Upsilon (\gcd(c,d)) \to \frac{1}{4}\,\frac{1}{\zeta(2)^2\,\mathbf{F}}, \quad\text{as $N \to \infty$}.$ and thus, from \ref{['eq:formula expect of ZMstar cuad']}, $\frac{A_N}{N^4}\to \frac{1}{\zeta(2)^2\,\mathbf{F}}, \quad\text{as $N \to \infty$}.$ Recalling the definition in \ref{['eq:formula correlacion']}, this gives \ref{['eq:average of correlations']}, as announced. We already know, see \ref{['eq:mean of ZMstar']}, that \mathbf{E}(Z^\star_M)^{2}=\frac{M^4}{\zeta(2)^{2}}\cdot We shall now use the results from the previous section to obtain that the variance of$Z^⋆_M$is$o(M^4)$, as$M ∞$. With the usual notations, $\lim_{M\to \infty}\frac{\mathbf{E}((Z^\star_M)^2)}{\mathbf{E}(Z^\star_M)^2}=1.$ and so, $\lim_{M\to \infty}\frac{\mathbf{V}(Z^\star_M)}{\mathbf{E}(Z^\star_M)^2}=0.$ Since $Z_M(a,b)=\sum_{(k,l) \in \mathcal{K}_M} V(a+k,b+l),$ we have that $Z_M(a,b)^2= \sum_{\substack{(k,l), (i,j)\in \mathcal{K}_M}} V(a+k,b+l)\,V(a+i, b+j) ,$ and thus, arguing as in \ref{['eq:mean of ZMstar']}, using \ref{['eq:precovariance']} and that $(k,l), (i,j)\in \mathcal{K}_M$ simply means that $1\le k,l,i,j\le M$, we deduce that \mathbf{E}((Z_M^\star)^2)=\lim_{n \to \infty} \mathbf{E}_n(Z_M^2) =\sum_{\substack{(k,l), (i,j)\in \mathcal{K}_M}} \mathbf{F} \,\Upsilon(\gcd(|i-k|, |j-l|))=\mathbf{F} \sum_{\substack{(k,i), (l,j)\in \mathcal{K}_M}} \Upsilon(\gcd(|i-k|, |j-l|)) =\mathbf{F}\, A_M, using the notation of \ref{['eq:def of AN']} in the last equality. Finally, thanks to \ref{['eq:limit AN']}, $\frac{\mathbf{E}((Z^\star_M)^2)}{\mathbf{E}(Z^\star_M)^2}=\frac{\zeta(2)^2}{M^4}\, \mathbf{F} A_M \to 1 , \quad \hbox{as $M \to \infty$}.\qedhere$ For$M 1$, let$Y_M^⋆$be the variable Y_M^\star=\frac{Z^\star_M}{M^2} which registers \emph{the average number} of coprime pairs in a random window of side length~$M$. We have that$E(Y^⋆_M)=1/ζ(2)$, see once more \ref{['eq:mean of ZMstar']}, and, because of Proposition \ref{['eq:big Oh varianza']}, that$_M ∞V(Y^⋆_M)=0$. Chebyshev's inequality gives immediately the following. The random variable $Y^\star_M$ tends, { in probability}, to the constant $1/\zeta(2)$ as $M\to \infty$. One could expect a result of asymptotic normality, as$M∞$, for a (convenient) normalization of the variable$Y_M^*$. For instance, one could consider the variables$U_M^⋆$,$M 1$, given by U_M^\star=M(Y_M^\star-\frac{1}{\zeta(2)}). (Observe that this normalization suggests that the variance of$Z_N^*$is of the order of$M^2$.) But, as observed by Sugita and Takanobu in SugitaTakanobu, and numerical experiments readily confirm, the behaviour of$U_M^⋆$may depend of the arithmetical properties of$M$. In Theorem~6 of~SugitaTakanobu, Sugita and Takanobu obtain a description of the limit points (not a unique one) of the sequence$(U_M^⋆)_M 1$in some$L^2$space of the adelic framework. Here are a few questions to understand further the peculiar dependence upon$M$of the distribution of$Z_M^⋆$or of$U_M^⋆$. (1) Is it the case that$V(Z_M^⋆)=O(M^2)$, with an absolute$O$, improving Proposition~\ref{['prop:OH bound for variance']}, and as suggested by Theorem 6 in~SugitaTakanobu? (2) Are the normalized variables$U_M^⋆$approximately a standard normal variable, for an appropriate sequence of sizes$M$tending to$∞$? (3) Recall Remark~\ref{['remark:ZMstar is poisson']}. Does the total variation distance between$Z_M^⋆$and a Poisson variable with parameter$M^2/ζ(2)$, depend upon the prime factorization of$M$? (4) Recall, from Remark \ref{['remark:caso Z=0']}, that for each$M$, there are$M$-windows which are fully "invisible": all points of the window have coordinates that are not coprime, that is,$Z_M(a,b)=0$. These invisible windows are rare, though. In the same vein as the previous question, one would expect$P(Z_M=0)$to be comparable to$e^-M^2/ζ(2)$. (5) A number of possible and natural extensions of the results of this paper could be explored. For instance, to higher dimensions, where one would have to distinguish between fully coprime tuples and pairwise coprime tuples (and also intermediate notions of coprimality, see Section 4 of FFracsam, or FFcodivisibility). And instead of the proportion of coprime pairs, one could consider the average gcd of the pairs in the random square or other moments of gcds within the square. Apostol, T.,M.: https://doi.org/10.1007/978-1-4757-5579-4. Undergrad. Texts Math., Springer, New York-Heidelberg, 1976.Cai, J.-Y. and Bach, E.: https://doi.org/10.1007/3-540-44679-6_53. In Computing and combinatorics COCOON 2001, pp. 473--482. Lecture Notes in Comput. Sci. 2108, Springer, Berlin, 2001.Ces`{a}ro, E.: https://doi.org/10.1007/bf02420800. Ann. Mat. Pura Appl. (2) 13 (1885), 233--268.Cilleruelo, J., Fern'andez, J.,L. and Fern'andez, P.: https://doi.org/10.1016/j.ejc.2018.08.004. European J. Combin. 75 (2019), 92--112.Dirichlet, P.,G.,L.: https://doi.org/10.1017/cbo9781139237345.007. In Abhandlungen der K"oniglich Preussischen Akademie der Wissenschaften con 1849, 69--83.Feller, W.: https://www.wiley.com/en-us/An+Introduction+to+Probability+Theory+and+Its+Applications,+Volume+1,+3rd+Edition-p-9780471257080. Third edition. John Wiley & Sons, New York-London-Sydney, 1968.Fern'andez, J.,L. and Fern'andez, P.: Equidistribution and coprimality. Preprint 2013, arXiv:,https://arxiv.org/abs/1310.3802.Fern'andez, J.,L. and Fern'andez, P.: {R}andom index of codivisibility. Preprint 2013, arXiv:,https://arxiv.org/abs/1310.4681.Fern'andez, J.,L. and Fern'andez, P.: https://doi.org/10.1007/s13398-020-00960-x. Rev. R. Acad. Cienc. Exactas F{'{}}s. Nat. Ser. A Mat. RACSAM 115 (2021), no. 1, article no. 26, 35 pp.Fern{' a}ndez, J.,L. and Fern{' a}ndez, P.: https://doi.org/10.37236/11424. Electron. J. Comb. 30 (2023), no. 2, article no. P2.11, 32 pp.Gerber, H.,U.: A proof of the Schuette--Nesbitt formula for dependent events. Actuarial Research Clearing House 1 (1979), 9--10.Gerber, H.,U.: Life insurance mathematics. Third edition, Springer, Berlin, 1997.Hardy, G.,H. and Wright, E.,M.: https://doi.org/10.1093/oso/9780199219858.001.0001. Sixth edition. Oxford University Press, Oxford, 2008.Herzog, F. and Stewart, B.,M.: https://doi.org/10.2307/2317753. Amer. Math. Monthly 78 (1971), 487--496.Martineau, S.: https://doi.org/10.1214/21-ecp381. Electron. Commun. Probab. 27 (2022), article no. 8, 14 pp.Sugita, H. and Takanobu, S.: https://projecteuclid.org/journals/osaka-journal-of-mathematics/volume-40/issue-4/The-probability-of-two-integers-to-be-co-prime-revisited/ojm/1153493406.full. Osaka J. Math. 40 (2003), no. 4, 945--976. $

Key Result

Lemma 2.1

Let $\{q_1, \ldots, q_R\}$ be a finite collection of distinct primes. Consider integers $u_1, \ldots, u_R$ and $v_1, \ldots, v_R$ such that $0\le u_j, v_j < q_j$, for $j=1,\dots,R$(which play the role of collections of residues). Let Then For $1\le S<R$, we have that

Figures (1)

  • Figure 1: The $M\times M$ window with base point $(a,b)$.

Theorems & Definitions (7)

  • Lemma 2.1
  • proof
  • Lemma 2.2: Schuette--Nesbitt
  • Lemma 2.3: Inclusion/exclusion principle and probability generating functions
  • proof
  • Lemma 2.4: Cesàro's identity
  • proof : Proof of Lemma \ref{['lemma:cesaro identity']}