The group structure for ζ(3)

Arithmetic of linear forms involving odd zeta values

“…The aim of this section is to prove two presentations of the linear form (2.7), (2.13): as a complex contour integral (in the spirit of [Gu,Ne1]) and as a real multiple integral (in the spirit of [Be,Ha5,RV3]). …”

“…We now recall that the family of linear forms in 1 and ζ(3) considered in paper [RV3] has the form (3.5)…”

“…Relations (3.8) and (3.10) enable us to describe the action of the generators ϕ, χ, ϑ, σ of the hypergeometric permutation group Φ from [RV3] in terms of the parameters (2.1):…”

“…A few months later, F. Beukers [Be] interpretated Apéry's sequence of rational approximations to ζ(2) and ζ(3) in terms of multiple integrals and Legendre polynomials. This approach was continued in later works [DV, Ru], [Ha1]- [Ha5], [HMV], [RV1]- [RV3] and yielded some new evaluations of the irrationality exponents for ζ(2), ζ(3), and other mathematical constants. Improvements of irrationality measures (i.e., upper bounds for irrationality exponents) for mathematical constants are closely related to another arithmetic approach, of eliminating extra prime numbers in binomials, introduced after G. V. Chudnovsky [Ch] by E. A. Rukhadze [Ru] and studied in detail by M. Hata [Ha1].…”

“…; see also [Ha1] for the explicit value of the constant on the right-hand side of (1.2). A further generalization of both the multiple integral approach and the arithmetic approach brings one to the group structures of G. Rhin and C. Viola [RV2,RV3]; their method yields the best known estimates for the irrationality exponents of ζ(2) and ζ(3):…”