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Automatic discovery of irrationality proofs and irrationality measures. (English) Zbl 1468.11160
The paper deals with the irrationality measure exponents of the numbers $$\zeta(3)$$, $$\zeta(2)$$, $$\log 2$$, the linear independence measure exponent of the numbers $$1$$, $$\log 2$$, $$\log 3$$ and other results concerning for example special integrals. They do some discussions and, with the help of several algorithms, they propose preliminary calculations leading to the estimations of irrationality and linear independence measure exponents.
##### MSC:
 11J82 Measures of irrationality and of transcendence 11J71 Distribution modulo one 11Y60 Evaluation of number-theoretic constants 33F10 Symbolic computation of special functions (Gosper and Zeilberger algorithms, etc.)
##### Software:
DEtools ; MultiZeilberger
Full Text:
##### References:
 [1] Alladi, K. and Robinson, M. L., Legendre polynomials and irrationality, J. Reine Angew. Math.318 (1980) 137-155. · Zbl 0425.10039 [2] Almkvist, G. and Berndt, B., Gauss, Landen, Ramanujan, the arithmetic-geometric mean, ellipses, pi and the Ladies Diary, Amer. Math. Mon.95 (1988) 585-608. · Zbl 0665.26007 [3] Almkvist, G. and Zeilberger, D., The method of differentiating under the integral sign, J. Symb. Comput.10 (1990) 571-591, http://www.math.rutgers.edu/zeilberg/mamarimY/duis.pdf. · Zbl 0717.33004 [4] Apagodu, M. and Zeilberger, D., Multi-variable Zeilberger and Almkvist-Zeilberger algorithms and the sharpening of Wilf-Zeilberger theory, Adv. Appl. Math.37 (2006) 139-152; http://sites.math.rutgers.edu/ zeilberg/mamarim/mamarimhtml/multiZ.html. · Zbl 1108.05010 [5] Beukers, F., A note on the irrationality of $$\zeta(2)$$ and $$\zeta(3)$$, Bull. London Math. Soc.11(3) (1979) 268-272. · Zbl 0421.10023 [6] Hata, M., The irrationality of $$\log(1+1/q)\log(1-1/q)$$, Trans. Amer. Math. Soc.350 (1998) 2311-2327. · Zbl 0918.11040 [7] Marcovecchio, R., The Rhin-Viola method for $$\log2$$, Acta Arith.139 (2009) 147-184. · Zbl 1197.11083 [8] M. Petkovsek, H. S. Wilf and D. Zeilberger, $$A=B$$ (A. K. Peters, 1996); https://www.math.upenn.edu/wilf/AeqB.html. [9] Rhin, G. and Viola, C., Linear independence of $$1, \operatorname{Li}_1$$ and $$\operatorname{Li}_2$$, Moscow J. Combin. Number Theory8(1) (2019) 81-96. · Zbl 1450.11074 [10] Rukhadze, E. A., A lower bound for the approximation of $$\ln2$$ by rational numbers, Vestnik Moskov. Univ. Ser. I Mat. Mekh. [Moscow Univ. Math. Bull.]42(6) (1987) 25-29(Russian). · Zbl 0635.10025 [11] Salikhov, V. Kh., On the irrationality measure of $$\ln3$$, Doklady Math.76 (2007) 955-957. · Zbl 1169.11032 [12] van der Poorten, A., A proof that Euler missed Apéry’s proof of the irrationality of $$\zeta(3)$$, An informal report, Math. Intell.1 (1979) 195-203; http://www.ega-math.narod.ru/Apery1.htm. · Zbl 0409.10028 [13] E. W. Weisstein, Irrationality Measure, from MathWorld — A Wolfram Web Resource; http://mathworld.wolfram.com/IrrationalityMeasure.html. [14] Zudilin, W., Difference equations and the irrationality measure of numbers, Proc. Steklov Inst. Math.218 (1997) 160-174. · Zbl 0910.11032 [15] Zudilin, W., An essay on irrationality measures of $$\pi$$ and other logarithms, Chebyshevskii Sb.5 (2004) 49-65(Russian); English version: http://arxiv.org/abs/math/0404523. · Zbl 1140.11036
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