Abstract
Using an integral of a hypergeometric function, we give necessary and sufficient conditions for irrationality of Euler’s constant γ. The proof is by reduction to known irrationality criteria for γ involving a Beukers-type double integral. We show that the hypergeometric and double integrals are equal by evaluating them. To do this, we introduce a construction of linear forms in 1, γ, and logarithms from Nesterenko-type series of rational functions. In the Appendix, S. Zlobin gives a change-of-variables proof that the series and the double integral are equal.
[1] BAILEY, W. N.: Generalised Hypergeometric Series, Cambridge University Press, Cambridge, 1935. Search in Google Scholar
[2] BEUKERS, F.: A note on the irrationality of ζ(2) and ζ(3), Bull. London Math. Soc. 11 (1979), 268–272. http://dx.doi.org/10.1112/blms/11.3.26810.1112/blms/11.3.268Search in Google Scholar
[3] GUILLERA, J.— SONDOW, J.: Double integrals and infinite products for some classical constants via analytic continuations of Lerch’s transcendent, Ramanujan J. 16 (2008), 247–270. http://dx.doi.org/10.1007/s11139-007-9102-010.1007/s11139-007-9102-0Search in Google Scholar
[4] HARDY, G. H.: Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, American Mathematical Society, Providence, 1999. Search in Google Scholar
[5] HESSAMI PILEHROOD, T.— HESSAMI PILEHROOD, KH.: Criteria for irrationality of generalized Euler’s constant, J. Number Theory 108 (2004), 169–185. http://dx.doi.org/10.1016/j.jnt.2004.05.00510.1016/j.jnt.2004.05.005Search in Google Scholar
[6] HUYLEBROUCK, D.: Similarities in irrationality proofs for π, ln 2, ζ(2), and ζ(3), Amer. Math. Monthly 108 (2001), 222–231. http://dx.doi.org/10.2307/269538310.2307/2695383Search in Google Scholar
[7] KRATTENTHALER, C.— RIVOAL, T.: How can we escape Thomae’s relations?, J. Math. Soc. Japan 58 (2006), 183–210. http://dx.doi.org/10.2969/jmsj/114528709810.2969/jmsj/1145287098Search in Google Scholar
[8] NESTERENKO, YU. V.: A few remarks on ζ(3), Math. Notes 59 (1996), 625–636. http://dx.doi.org/10.1007/BF0230721210.1007/BF02307212Search in Google Scholar
[9] NESTERENKO, YU. V.: Integral identities and constructions of approximations to zeta-values (Actes des 12èmes Rencontres Arithmétiques de Caen (June 2001)), J. Théor. Nombres Bordeaux 15 (2003), 535–550. 10.5802/jtnb.412Search in Google Scholar
[10] PRÉVOST, M.: A family of criteria for irrationality of Euler’s constant, http://arxiv.org/abs/math/0507231 (2005). Search in Google Scholar
[11] SONDOW, J.: A hypergeometric approach, via linear forms involving logarithms, to irrationality criteria for Euler’s constant, with an Appendix by Sergey Zlobin, http://arxiv.org/abs/math/0211075v2 (2002). Search in Google Scholar
[12] SONDOW, J.: Criteria for irrationality of Euler’s constant, Proc. Amer. Math. Soc. 131 (2003), 3335–3344. http://dx.doi.org/10.1090/S0002-9939-03-07081-310.1090/S0002-9939-03-07081-3Search in Google Scholar
[13] SONDOW, J.: An infinite product for e γvia hypergeometric formulas for Euler’s constant, γ, http://arXiv.org/abs/math/0306008 (2003). Search in Google Scholar
[14] SONDOW, J.: Double integrals for Euler’s constant and ln 4/π and an analog of Hadjicostas’s formula, Amer. Math. Monthly 112 (2005), 61–65. Search in Google Scholar
[15] SONDOW, J.: A faster product for π and a new integral for ln π/2, Amer. Math. Monthly 112 (2005), 729–734. http://dx.doi.org/10.2307/3003757510.2307/30037575Search in Google Scholar
[16] SONDOW, J.— HADJICOSTAS, P.: The generalized-Euler-constant function γ(z) and a generalization of Somos’s quadratic recurrence constant, J. Math. Anal. Appl. 332 (2007), 292–314. http://dx.doi.org/10.1016/j.jmaa.2006.09.08110.1016/j.jmaa.2006.09.081Search in Google Scholar
[17] SONDOW, J.— ZUDILIN, W.: Euler’s constant, q-logarithms, and formulas of Ramanujan and Gosper, Ramanujan J. 12 (2006), 225–244. http://dx.doi.org/10.1007/s11139-006-0075-110.1007/s11139-006-0075-1Search in Google Scholar
[18] ZUDILIN, W.: A few remarks on linear forms involving Catalan’s constant, Chebyshevskii Sb. 3 (2002) no. 2(4), 60–70 (Russian); http://arXiv.org/abs/math/0210423 (English). Search in Google Scholar
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