The number $\pi$ and summation by $SL(2,\mathbb Z)$

Let $f(a,b,c,d)=\sqrt{a^2+b^2}+\sqrt{c^2+d^2}-\sqrt{(a+c)^2+(b+d)^2}$. (it is the defect in the triangle inequality)

Then, we discovered by heuristic arguments and then verified by computer that $$\sum f(a,b,c,d)^n = 2-\pi/2$$ where the sum runs over all $a,b,c,d\in\mathbb Z$ such that $a\geq 1,b,c\geq 0, ad-bc=1$ and $n=2$.

It seems that when $n=1$, we obtain $2$ in the right hand side. We have failed to guess the result for $n>2$.

So the question is: can you prove the result for $n=2$? (We can, but in a rather unnatural way. We will write this later and now we want to tease the community, probably somebody can find a beautiful proof.)

Added: we have two answers for the above question, so the rest is:

Question: Can you guess the result for $n>2$? I tried http://mrob.com/pub/ries/ but nothing interesting was revealed.

PS. In case it can help someone, below are these sums of powers ($n=1,2,3,4,5$), calculated by computer:

$1.9955289122768913 = 2$

$0.4292036731309361 = 2 - \pi/2$

$0.21349025954227965 =$ ?

$0.11983665032283052 =$ ?

$0.06933955916793563 =$ ?

Added: this and something more can be found in https://arxiv.org/abs/1701.07584 and https://arxiv.org/abs/1711.02089

• I was so struck by your lovely identities for n = 1,2 that I had to do numerical evaluations of my own to see this in action. Wow! – Steve Huntsman Sep 16 '16 at 20:23
• I wonder if this problem can be attacked by the spectral theory of $\mathbb{H}/\textrm{SL}(2,\mathbb{Z})$. If we sum over all of $\textrm{SL}(2,\textrm{Z})$, we get $F(e)$ where $F(g)$ is the corresponding sum over bases of the lattice $g\mathbb{Z}^2$, giving an $SO(2)$-invariant function on the space of lattices, which therefore descends to the modular curve. Now if the resulting function is in $L^2$ and orthogonal to the cuspidal space, we can write $F(e)$ as an integral against Eisenstein series and perhaps use some complex analysis tricks to evaluate it. Does this sound plausible? – guest Sep 16 '16 at 21:01
• @guest here we sum over only "a half" of $SL(2,\mathbb Z)$, it is not clear to me how to rewrite the sum in order to use all elements of the group. You see, in order to have convergence, we consider only triangles with area 1 and the longest diagonal passing through $(0,0)$ – Nikita Kalinin Sep 16 '16 at 21:19
• @AndreiSmolensky: I would tell this is just the effect of the raising role of the maximal value of $f$ which is $f(1,0,0,1)=2-\sqrt2=\bigl(1+\frac{\sqrt2}2\bigr)^{-1}$. – Ilya Bogdanov Sep 17 '16 at 23:58
• @SamHopkins There is a probabilistic sense for $n=2$. See my answer for this question. There is a map from points in the square minus disk to $SL(2,\mathbb Z)$, and $f(a,b,c,d)^2/2$ is the measure of the preimage of this map for $(a,b,c,d)\in SL(2,\mathbb Z)$. – Nikita Kalinin Sep 18 '16 at 2:29

Here goes $n=2$ a la Fedor Petrov.

Notice that his argument is based on the smart identities $x_z+y_z=|z|$, $z_x=|x|+y_x$ and $z_y=|y|+x_y$ where $x,y$ are 2 vectors in $\mathbb R^2$, $z=x+y$ and $x_y$ is the (signed in general, but in our setting everything is non-negative) length of the projection of $x$ to $y$. Fedor's idea can be made into a one-liner: consider $\Phi_1(x,y)=|x|+|y|-x_y-y_x$, check the identity $\Phi_1(x,y)-\Phi_1(z,x)-\Phi_1(z,y)=|x|+|y|-|z|$ and telescope.

For $n=2$ the right function to consider is $$\Phi_2(x,y)=2|x||y|-|x|y_x-|y|x_y$$ I wrote it in the symmetric form, but actually $|x|y_x=|y|x_y=|x||y|\cos\theta$ where $\theta$ is the angle between $x$ and $y$. The difficulty is that in this case the decay "at infinity" is not fast enough to send the sum of faraway terms to $0$: the typical term there is $2|x||y|(1-\cos\theta)=(2|x||y|\sin\frac{\theta }2)2\sin\frac{\theta}2\approx A\theta$ where $A$ is the (invariant) area of the parallelogram spanned by $x$ and $y$. The sum of these expressions at infinity adds an extra term $A$ times the angle between $x$ and $y$ with $-$ sign (the precision is now good enough to ignore the rest; like Fedor, I'll leave the "routine convergence checks" to the reader). So, for two perpendicular unit vectors, we get $2\cdot1\cdot1-0-0-1\cdot\frac \pi 2$ as requested.

I wonder if we can continue with that a bit. The first task would be to find $\Phi_3$ but, even if we are lucky and it exists, it may require even more correcting terms when trying to telescope and we can get a transcendental problem there.

Let me write down here a proof that $\sum f(a,b,c,d)=2$, maybe someone sees how this may be generalized for the second moment. I do not.

We denote the vectors $x=(a,b)$ and $y=(c,d)$ and write $g(x,y)=f(a,b,c,d)$. Denote $S=\sum g(x,y)$, I omit here a routine proof that $S$ is finite. We may fix $x$ and sum up by $y$. For given $x$ the possible $y$ have form $y_0(x)+kx$ for $k=0,1,2,\dots$. The sum by all these $y$ equals $$\lim_N \sum_{k=0}^{N-1} \|x\|+\|y_0(x)+kx\|-\|y_0(x)+(k+1)x\|=\\ =\lim_N N\|x\|+\|y_0(x)\|-\|y_0+Nx\|=\|y_0(x)\|-pr_x (y_0(x)),$$ where $pr_x$ denotes projection onto $x$. So, $S=\sum_x \|y_0(x)\|-pr_x (y_0(x)).$ Analogously we may fix $y$ and sum up by $x$, and get a similar formula $S=\sum_y \|x_0(y)\|-pr_y (x_0(y))$. In the first formula we take a summand corresponding to $x=(1,0)$ and in the second a summand corresponding to $y=(0,1)$. Those summands are equal to 1. Other summands correspond to $x$, resp. $y$, with strictly positive coordinates. Any such vector, which I denote by $z$, is a sum of $x_0(z)$ and $y_0(z)$, thus when we sum up two expressions for $S$ we get $$2S=1+1+\sum_z \|x_0(z)\|+\|y_0(z)\|-\|z\|=2+S,$$ and $S=2$.

I add our explanation and the origin of the problem.

To obtain the formula we just need to verify the following lemma (by a straightforward computation).

Magic Lemma. Let $(a,b),(c,d)$ be as in our summation in the main posting. Draw tangents to the unit circle which are orthogonal to the directions $(a,b),(c,d),(a+c,b+d)$. Then the triangle in the intersection has the area $\frac{f(a,b,c,d)^2}{2}$.

Now consider the circle given by $x^2+y^2=1$. At every rational point on it we draw the tangent line. Clearly, our disk is the intersection of all the half-planes given by these lines.

So we can approximate the area of the disk by the following sequential cutting. We start with the square $[-1,1]\times [-1,1]$. It has four points of tangency with our circle. We will construct a sequence of polygons converging to the circle. Initially we have sides of directions orthogonal to $(1,0),(0,1),(-1,0),(0,1)$. At every step, we take two vectors $v_1,v_2$(in the same quadrant) which give the basis of $\mathbb Z^2$ and add to our polygon a new side, which is given by the tangent line of the direction orthogonal to $v_1+v_2$.

For example, the first step will be cutting the square by the line $x+y=\sqrt 2$, if we take the vectors $(1,0),(0,1)$. The area of the eliminated triangle is $\frac{(\sqrt2-1)^2}{2}=\frac{f(1,0,0,1)^2}{2}$.

It seems that this procedure can be generalized to higher dimensions — at least a preliminary computation shows that something like the above magic lemma should happen.

• I am afraid that in higher dimension the simplices you cut off may not form a partition of the region between a ball and a cube. – Fedor Petrov Sep 18 '16 at 10:03
• @FedorPetrov yes, indeed. And so we need to have corrections.Kind of inclusion-exclusion formula. I hope. – Nikita Kalinin Sep 18 '16 at 14:07
• Why do you call it "a rather unnatural way"? It looks very natural. – Alexey Ustinov Sep 20 '16 at 20:55