The Number π and a Summation by SL(2 , Z)

Nikita Kalinin*, Mikhail Shkolnikov

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

2 Scopus citations


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)=a2+b2+c2+d2-(a+c)2+(b+d)2. Then, [Figure not available: see fulltext.][Figure not available: see fulltext.] where the sum runs by all a, b, c, d∈ Z≥ 0 such that ad- bc= 1. We present a proof of these formulae and list several directions for the future studies.

Original languageEnglish
Pages (from-to)511-517
Number of pages7
JournalArnold Mathematical Journal
Issue number4
StatePublished - 1 Dec 2017
Externally publishedYes


  • Lattice geometry
  • Special linear group
  • Summation
  • Tropical geometry
  • pi


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