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Arithmetic of linear forms involving odd zeta values. (English) Zbl 1156.11327
From the introduction: The aim of this paper is to explain the group structures used for evaluating the irrationality exponents \[ \mu(\log 2)\leq 3.891399 . . . \quad \mu(\zeta(2)) \leq 5.441242 . . . ,\quad\mu(\zeta(3)) \leq 5.513890 . . . , \] via Nesterenko’s method, as well as to present a new result on the irrationality of the odd zeta values inspired by Rivoal’s construction and possible generalizations of the Rhin-Viola approach.
This paper is organized as follows. In Sections 2–5 we explain in detail the group structure of Rhin and Viola for \(\zeta(3)\); we do not use Beukers’ type integrals as in [G. Rhin and C. Viola, Acta Arith. 97, No. 3, 269–293 (2001; Zbl 1004.11042)] for this, but with the use of Nesterenko’s theorem we explain all stages of our construction in terms of their doubles from [loc. cit.].
Section 6 gives a brief overview of the group structure for \(\zeta(2)\) from [G. Rhin and C. Viola, Acta Arith. 77, No. 1, 23–56 (1996; Zbl 0864.11037)]. Section 7 is devoted to a study of the arithmetic of rational functions appearing naturally as ‘bricks’ of general Nesterenko’s construction [Yu. V. Nesterenko, Arithmetic properties of values of the Riemann zeta function and generalized hypergeometric functions. Manuscript (2001)].
In Section 8 we explain the well-poised hypergeometric origin of Rivoal’s construction and improve the previous result from [T. Rivoal, Acta Arith. 103, No. 2, 157–167 (2002; Zbl 1015.11033)], the author, Math. Notes 70, No. 3, 426–431 (2001); translation from Mat. Zametki 70, No. 3, 472–476 (2001; Zbl 1022.11035)] on the irrationality of \(\zeta(5)\), \(\zeta(7), \dots\); namely, we state that at least one of the four numbers
\[ \zeta(5), \zeta(7), \zeta(9),\;\text{and}\;\zeta(11)\ \text{is irrational}. \]
Although the success of our new result from Section 8 is due to the arithmetic approach, in Section 9 we present possible group structures for linear forms in 1 and odd zeta values; these groups may become useful, provided that some arithmetic condition (which we indicate explicitly) holds.

MSC:
11J72 Irrationality; linear independence over a field
11M06 \(\zeta (s)\) and \(L(s, \chi)\)
33D15 Basic hypergeometric functions in one variable, \({}_r\phi_s\)
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