On Catalan's constant

Yu. V. Nesterenko

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Algebra, geometry, and number theory, Tome 292 (2016), pp. 159-176.

Résumé

A new efficient construction of Diophantine approximations to Catalan's constant is presented that is based on the direct analysis of the representation of a hypergeometric function with specially chosen half-integer parameters as a series and as a double Euler integral over the unit cube. This allows one to significantly simplify the proofs of Diophantine results available in this domain and substantially extend the capabilities of the method. The sequences of constructed rational approximations are not good enough to prove irrationality, but the results established allow one to compare the quality of various constructions.

Bibliographie

[1] Aptekarev A.I., Lysov V.G., “Asimptotika $\gamma $-form, generiruemykh sovmestno ortogonalnymi mnogochlenami”, Ratsionalnye priblizheniya postoyannoi Eilera i rekurrentnye sootnosheniya, Sovr. probl. matematiki, 9, MIAN, M., 2007, 55–62 | DOI

[2] Berndt B.C., Ramanujan's Notebooks, Part 1, Springer, New York, 1985 | MR | Zbl

[3] Catalan E., Mémoire sur la transformation des séries et sur quelques intégrales définies, Mém. Acad. r. sci. Belgique, 33, Hayez, Bruxelles, 1865

[4] Catalan E., Recherches sur la constante $G$, et sur les intégrales eulériennes, Mém. Acad. impér. sci. St. Petersbourg. Sér. 7, 31, no. 3, St. Petersbourg, 1883

[5] Feldman N.I., Sedmaya problema Gilberta, Izd-vo MGU, M., 1982 | MR

[6] Hessami Pilehrood Kh., Hessami Pilehrood T., “On a continued fraction expansion for Euler's constant”, J. Number Theory, 133:2 (2013), 769–786 | DOI | MR | Zbl

[7] Khessami Pilerud T.G., “Otsenki snizu lineinykh form ot znachenii nekotorykh gipergeometricheskikh funktsii”, Mat. zametki, 67:3 (2000), 441–451 | DOI | MR

[8] Ivankov P.L., “Ob arifmeticheskikh svoistvakh znachenii gipergeometricheskikh funktsii”, Mat. sb., 182:2 (1991), 283–302 | MR

[9] Krattenthaler C., Rivoal T., “On a linear form for Catalan's constant”, South East Asian J. Math. Sci., 6:2 (2008), 3–15 | MR | Zbl

[10] Lyuk Yu., Spetsialnye matematicheskie funktsii i ikh approksimatsii, Mir, M., 1980

[11] Matveev E.M., “Ob arifmeticheskikh svoistvakh znachenii obobschennykh binomialnykh mnogochlenov”, Mat. zametki, 54:4 (1993), 76–81 | MR | Zbl

[12] Nesterenko Yu.V., “O pokazatele irratsionalnosti chisla $\ln 2$”, Mat. zametki, 88:4 (2010), 549–564 | DOI | MR | Zbl

[13] Nielsen N., “Der Eulersche Dilogarithmus und Seine Verallgemeinerungen: Eine Monographie”, Nova acta Leop., 90 (1909), 121–212

[14] Rivoal T., “Nombres d'Euler, approximants de Padé et constante de Catalan”, Ramanujan J., 11:2 (2006), 199–214 | DOI | MR | Zbl

[15] Rivoal T., “Rational approximations for values of derivatives of the Gamma function”, Trans. Amer. Math. Soc., 361:11 (2009), 6115–6149 | DOI | MR | Zbl

[16] Rivoal T., Zudilin W., “Diophantine properties of numbers related to Catalan's constant”, Math. Ann., 326:4 (2003), 705–721 | DOI | MR | Zbl

[17] Uitekker E.T., Vatson Dzh.N., Kurs sovremennogo analiza, Ch. 2, Fizmatgiz, M., 1963

[18] Zudilin V.V., “Nekotorye zamechaniya o lineinykh formakh, soderzhaschikh postoyannuyu Katalana”, Chebyshev. sb., 3:2 (2002), 60–70 | MR | Zbl

[19] Zudilin V.V., “O rekursii tretego poryadka tipa Aperi dlya $\zeta (5)$”, Mat. zametki, 72:5 (2002), 796–800 | DOI | MR | Zbl

[20] Zudilin W., “An Apéry-like difference equation for Catalan's constant”, Electron. J. Comb., 10:1 (2003), Pap. R14 | MR