The Number $$\pi $$π and a Summation by $$SL(2,{\mathbb {Z}})$$SL(2,Z)

  title={The Number \$\$\pi \$\$π and a Summation by \$\$SL(2,\{\mathbb \{Z\}\})\$\$SL(2,Z)},
  author={Nikita Kalinin and Mikhail Shkolnikov},
  journal={Arnold Mathematical Journal},
  • Nikita Kalinin, Mikhail Shkolnikov
  • Published 2017
  • Mathematics, Physics
  • Arnold Mathematical Journal
  • 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 $$f(a,b,c,d)=\sqrt{a^2+b^2}+\sqrt{c^2+d^2}-\sqrt{(a+c)^2+(b+d)^2}$$f(a,b,c,d)=a2+b2+c2+d2-(a+c)2+(b+d)2. Then, where the sum runs by all $$a,b,c,d\in {\mathbb {Z}}_{\ge 0}$$a,b,c,d∈Z≥0 such that $$ad-bc=1$$ad-bc=1. We present a proof of these formulae and list several directions for the future studies. 
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