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Tropical formulae for summation over a part of SL(2,Z)
European Journal of Mathematics. 2019. Vol. 5. No. 3. P. 909–928.
Shkolnikov M., Kalinin N.
√√
Abstract Let f(a,b,c,d) = a2+b2 + c2+d2 − (a+c)2+(b+d)2, let
(a,b,c,d) stand for a,b,c,d ∈ Z0 such that ad − bc = 1. Defines
In other words, we consider the sum of the powers of the triangle inequality defects for the lattice parallelograms (in the first quadrant) of area one.
We prove that F(s) converges when s > 1 and diverges at s = 1/2. We also prove that
1=1,(a,b,c,d) (a+c)2(b+d)2(a+b+c+d)2 3
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