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A few remarks on $$\zeta (3)$$. (English. Russian original) Zbl 0888.11028
Math. Notes 59, No. 6, 625-636 (1996); translation from Mat. Zametki 59, No. 6, 865-880 (1996).
This is an exciting paper packed with wonderful formulas, valuable analysis, and useful explanations, which will surely interest the many ‘fans’ of Apéry’s astonishing proof in 1978 of the irrationality of $$\zeta(3)$$ (see the reviewer’s ‘A proof that Euler missed …’ [A. J. van der Poorten, Math. Intell. 1, 195-203 (1979; Zbl 0409.10028)]. Specifically, the author analyses and applies ideas of L. A. Gutnik [Acta Arith. 42, 255-264 (1983; Zbl 0521.10028)] to produce a new view on Apéry’s proof. These notions give rise to the author’s extensive exploration of recurrence relations satisfied by a class of Meyer $$G$$-functions and lead to new continued fraction representations of $$\zeta(3)$$. This paper gives one a fine response to provide to those writing to ask “what has happened to $$\zeta(3)$$ since 1979?”.

##### MSC:
 11J72 Irrationality; linear independence over a field 11M06 $$\zeta (s)$$ and $$L(s, \chi)$$ 11J70 Continued fractions and generalizations
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##### References:
 [1] R. Apery, ”Irrationalité de $$\zeta$$(2) et $$\zeta$$(3),”Astérisque,61, 11–13 (1979). [2] H. Cohen, ”Démonstration de l’irrationalité de $$\zeta$$(3) (d’après R. Apery),” in:Semin. de théorie des nombres, Grenoble (1978). [3] A. van der Poorten, ”A proof that Euler missed–Apery’s proof of the irrationality of $$\zeta$$(3),”Math. Intell.,1, 195–203 (1979). · Zbl 0409.10028 · doi:10.1007/BF03028234 [4] E. Reyssat, ”Irrationalité de $$\zeta$$(3), selon Apery,” in:20 année. Seminaire Delange-Pisot-Poitou, (1978/79), p. 6. [5] F. Beukers, ”A note on the irrationality of $$\zeta$$(2) and $$\zeta$$(3),”Bull. London Math. Soc.,11, 268–272 (1979). · Zbl 0421.10023 · doi:10.1112/blms/11.3.268 [6] M. Hata, ”On the linear independence of the values of polylogarithmic functions,”J. math. pures et appl.,69, 133–173 (1990). · Zbl 0712.11040 [7] L. A. Gutnik, ”On the irrationality of certain quantities involving $$\zeta$$(3),”Uspekhi Mat. Nauk [Russian Math. Surveys],34, No. 3, 190 (1979). Translation in:Acta Arith.,42 No. 3, 255–264 (1983). · Zbl 0474.10026 [8] Y. L. Luke,Mathematical Functions and Their Approximations, Academic Press, New York-San Francisko-London (1975). · Zbl 0318.33001 [9] F. Beukers, ”Pade approximations in number theory,” in:Lecture Notes in Math, Vol. 888, Springer-Verlag, New York-Berlin (1981), pp. 90–99. [10] V. N. Sorokin, ”Hermite-Pade approximations for Nikishin systems and the irrationality of $$\zeta$$(3),”Uspekhi Mat. Nauk [Russian Math. Surveys],49, No. 2, 167–168 (1994). [11] W. B. Jones and W. J. Thron,Continued Fractions. Analytic Theory and Applications, Encyclopedia of Mathematics and Its Applications, Vol. 11, Addison-Wesley, Reading, Mass. (1980). · Zbl 0445.30003 [12] A. O. Gelfond,Calculus of Finite Differences [in Russian], Nauka, Moscow (1967).
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