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

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Abstract

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.

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References

  • Cauchy, A.L.: Note VIII. Cours d’Analyse de l’École Royale Polytechnique; I.re Partie. Analyse algébrique (1821)

  • Euler, L.: De summis serierum reciprocarum. Commentarii academiae scientiarum Petropolitanae 7, 123–134 (1740). (E41 in the Eneström index. An English translation can be found in arXiv:math/0506415v2)

  • Kalinin, N., Shkolnikov, M.: Tropical curves in sandpiles. Comptes Rendus Mathematique 354(2), 125–130 (2016)

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  • Kalinin, N., Shkolnikov, M.: Introduction to tropical series and wave dynamic on them (2017). arXiv:1706.03062

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Acknowledgements

We want to thank God and the universe for these beautiful formulae. Also we would like to thank an anonymous referee for the idea to discuss the Euler formula, and Fedor Petrov and Pavol Ševera for fruitful discussions. The first author, Nikita Kalinin, is funded by the SNSF PostDoc.Mobility Grant 168647 and thanks Grant FORDECYT-265667 “Programa para un avance global e integrado de la matemática mexicana”. Also, support from the Basic Research Program of the National Research University Higher School of Economics is gratefully acknowledged. The second author, Mikhail Shkolnikov is partially supported by the Grant 159240 of the Swiss National Science Foundation as well as by the National Center of Competence in Research SwissMAP of the Swiss National Science Foundation.

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Authors and Affiliations

  1. National Research University Higher School of Economics, Soyuza Pechatnikov Str., 16, St. Petersburg, Russian Federation

    Nikita Kalinin

  2. Section de Mathématiques, University of Geneva, Route de Drize 7, Villa Battelle, 1227, Carouge, Switzerland

    Mikhail Shkolnikov

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  1. Nikita Kalinin
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  2. Mikhail Shkolnikov
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Correspondence to Nikita Kalinin.

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Kalinin, N., Shkolnikov, M. The Number \(\pi \) and a Summation by \(SL(2,{\mathbb {Z}})\) . Arnold Math J. 3, 511–517 (2017). https://doi.org/10.1007/s40598-017-0075-9

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  • DOI: https://doi.org/10.1007/s40598-017-0075-9

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