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Article

The number π and summation by SL(2, Z)

Arnold Mathematical Journal. 2017. Vol. 3. No. 4. P. 511-517.
Kalinin N., Shkolnikov M.

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, 

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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.