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The Number π and a Summation by SL(2 , Z). / Kalinin, Nikita; Shkolnikov, Mikhail.

In: Arnold Mathematical Journal, Vol. 3, No. 4, 01.12.2017, p. 511-517.

Research output: Contribution to journal › Article › peer-review

Harvard

Kalinin, N & Shkolnikov, M 2017, 'The Number π and a Summation by SL(2 , Z)', Arnold Mathematical Journal, vol. 3, no. 4, pp. 511-517. https://doi.org/10.1007/s40598-017-0075-9

APA

Kalinin, N., & Shkolnikov, M. (2017). The Number π and a Summation by SL(2 , Z). Arnold Mathematical Journal, 3(4), 511-517. https://doi.org/10.1007/s40598-017-0075-9

Vancouver

Kalinin N, Shkolnikov M. The Number π and a Summation by SL(2 , Z). Arnold Mathematical Journal. 2017 Dec 1;3(4):511-517. https://doi.org/10.1007/s40598-017-0075-9

Author

Kalinin, Nikita ; Shkolnikov, Mikhail. / The Number π and a Summation by SL(2 , Z). In: Arnold Mathematical Journal. 2017 ; Vol. 3, No. 4. pp. 511-517.

BibTeX

@article{6c0d3c49c1b24cd2ac4cae6586ba526f,
title = "The Number π and a Summation by SL(2 , Z)",
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)=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.",
keywords = "Lattice geometry, pi, Special linear group, Summation, Tropical geometry",
author = "Nikita Kalinin and Mikhail Shkolnikov",
year = "2017",
month = dec,
day = "1",
doi = "10.1007/s40598-017-0075-9",
language = "English",
volume = "3",
pages = "511--517",
journal = "Arnold Mathematical Journal",
issn = "2199-6792",
publisher = "Springer Nature",
number = "4",

}

RIS

TY - JOUR

T1 - The Number π and a Summation by SL(2 , Z)

AU - Kalinin, Nikita

AU - Shkolnikov, Mikhail

PY - 2017/12/1

Y1 - 2017/12/1

N2 - 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.

AB - 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.

KW - Lattice geometry

KW - pi

KW - Special linear group

KW - Summation

KW - Tropical geometry

UR - http://www.scopus.com/inward/record.url?scp=85033669876&partnerID=8YFLogxK

U2 - 10.1007/s40598-017-0075-9

DO - 10.1007/s40598-017-0075-9

M3 - Article

AN - SCOPUS:85033669876

VL - 3

SP - 511

EP - 517

JO - Arnold Mathematical Journal

JF - Arnold Mathematical Journal

SN - 2199-6792

IS - 4

ER -

ID: 49793573