The Number $$\pi $$ π and a Summation by $$SL(2,{\mathbb {Z}})$$ S L ( 2 , Z )
Nikita Kalinin, Mikhail Shkolnikov · Arnold Mathematical Journal · 2017
The sum (resp. the sum of squares) of the defects in the triangle inequalities for the area one lattice parallelograms in the first quadrant has a surprisingly simple expression. Namely, let $$f(a,b,c,d)=\sqrt{a^2+b^2}+\sqrt{c^2+d^2}-\sqrt{(a+c)^2+(b+d)^2}$$ . Then, where the sum runs by all $$a,b,c,d\in {\mathbb {Z}}_{\ge 0}$$ such that $$ad-bc=1$$ . We present a proof of these formulae and list several directions for the future studies.