The question The number $\pi$ and summation by $SL(2,\mathbb Z)$ considers the series $$\sum(\lVert x\rVert+\lVert y\rVert-\lVert x+y\rVert)$$ where the series runs over vectors $x,y\in{\mathbb N}^2$ such that $\det(x,y)=1$. The norm being the standard euclidean one. Its value (equal to $2$) is obtained by a telescoping argument. However, the telescopage requires the knowledge that the auxiliary series $$\sum\frac1{\lVert x\rVert^2\lVert y\rVert}$$ converges. For this, we need to estimate the distribution of matrices in $\operatorname{SL}_2({\mathbb N})$, in terms of the lengths of their columns.

What is known in this direction? What is approximately the number of matrices $M=(x,y)\in \operatorname{SL}_2({\mathbb N})$ such that $\lVert x\rVert+\lVert y\rVert\le R$, as $R\rightarrow+\infty$?

The question The number $\pi$ and summation by $SL(2,\mathbb Z)$ considers the series $$\sum(\lVert x\rVert+\lVert y\rVert-\lVert x+y\rVert)$$ where the series runs over vectors $x,y\in{\mathbb N}^2$ such that $\det(x,y)=1$. The norm being the standard euclidean one. Its value (equal to $2$) is obtained by a telescoping argument. However, the telescopage requires the knowledge that the auxiliary series $$\sum\frac1{\lVert x\rVert\lVert y\rVert(\lVert x\rVert+\lVert y\rVert)}$$ converges. For this, we need to estimate the distribution of matrices in $\operatorname{SL}_2({\mathbb N})$, in terms of the lengths of their columns.

What is known in this direction? What is approximately the number of matrices $M=(x,y)\in \operatorname{SL}_2({\mathbb N})$ such that $\lVert x\rVert+\lVert y\rVert\le R$, as $R\rightarrow+\infty$?

This question considers the series $$\sum(\|x\|+\|y\|-\|x+y\|)$$ where the series runs over vectors $x,y\in{\mathbb N}^2$ such that $\det(x,y)=1$. The norm being the standard euclidian one. Its value (equal to $2$) is obtained by a telescoping argument. However, the telescopage requires the knowledge that the auxiliary series $$\sum\frac1{\|x\|^2\|y\|}$$ converges. For this, we need to estimate the distribution of matrices in $SL_2({\mathbb N})$, in terms of the lengths of their columns.

What is known in this direction  ? What is approximately the number of matrices $M=(x,y)\in \operatorname{SL}_2({\mathbb N})$ such that $\|x\|+\|y\|\le R$, as $R\rightarrow+\infty$  ?

The question The number $\pi$ and summation by $SL(2,\mathbb Z)$ considers the series $$\sum(\lVert x\rVert+\lVert y\rVert-\lVert x+y\rVert)$$ where the series runs over vectors $x,y\in{\mathbb N}^2$ such that $\det(x,y)=1$. The norm being the standard euclidean one. Its value (equal to $2$) is obtained by a telescoping argument. However, the telescopage requires the knowledge that the auxiliary series $$\sum\frac1{\lVert x\rVert^2\lVert y\rVert}$$ converges. For this, we need to estimate the distribution of matrices in $\operatorname{SL}_2({\mathbb N})$, in terms of the lengths of their columns.

What is known in this direction? What is approximately the number of matrices $M=(x,y)\in \operatorname{SL}_2({\mathbb N})$ such that $\lVert x\rVert+\lVert y\rVert\le R$, as $R\rightarrow+\infty$?

This question considers the series $$\sum(\|x\|+\|y\|-\|x+y\|)$$ where the series runs over vectors $x,y\in{\mathbb N}^2$ such that $\det(x,y)=1$. The norm being the standard euclidian one. Its value (equal to $2$) is obtained by a telescoping argument. However, the telescopage requires the knowledge that the auxiliary series $$\sum\frac1{\|x\|^2\|y\|}$$ converges. For this, we need to estimate the distribution of matrices in $SL_2({\mathbb N})$, in terms of the lengths of their columns.

What is known in this direction ? What is approximately the number of matrices $M=(x,y)\in SL_2({\mathbb N})$ such that $\|x\|+\|y\|\le R$, as $R\rightarrow+\infty$ ?

This question considers the series $$\sum(\|x\|+\|y\|-\|x+y\|)$$ where the series runs over vectors $x,y\in{\mathbb N}^2$ such that $\det(x,y)=1$. The norm being the standard euclidian one. Its value (equal to $2$) is obtained by a telescoping argument. However, the telescopage requires the knowledge that the auxiliary series $$\sum\frac1{\|x\|^2\|y\|}$$ converges. For this, we need to estimate the distribution of matrices in $SL_2({\mathbb N})$, in terms of the lengths of their columns.

What is known in this direction ? What is approximately the number of matrices $M=(x,y)\in \operatorname{SL}_2({\mathbb N})$ such that $\|x\|+\|y\|\le R$, as $R\rightarrow+\infty$ ?