On a permutation group related to ζ(2)

Heine's basic transform and a permutation group for q-harmonic series

“…The series (13) involving linear forms in 1 and the q-harmonic series can be represented as some q-integrals (see, e.g., [Ex,Section 2.5.1]). This q-integral representation is very similar to that used in [RV1] and [RV2] for describing the permutation groups for ζ(2) and ζ(3). In spite of this similarity, there exists no general pattern to change the variable of q-integration (see [Ask] and [Ex,Section 2.2.4]).…”

“…The "ordinary" arithmetic approach occurs as a part of the groupstructure approach proposed by G. Rhin and C. Viola in [RV1], [RV2] for obtaining quantitative results for the values ζ(2) and ζ(3) of Riemann's zeta function. Recently, the author [Zu2] extended the method of [RV1] to a suitable q-analogue of ζ(2).…”

“…Recently, the author [Zu2] extended the method of [RV1] to a suitable q-analogue of ζ(2). The permutation group in [RV1], [Zu2] is rather rich to get nice estimates for irrationality exponents in both ordinary and q-(basic) cases. A simpler group (of order 12) for the q-harmonic series that appears below also leads to a quantitative result.…”

Arithmetic of linear forms involving odd zeta values

“…hence we can identify the quantity (6.3) with the corresponding integral I(h, i, j, k, l) from [RV2] by setting…”

“…; 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):…”