Multiple integrals and linear forms in zeta-values
نویسندگان
چکیده
منابع مشابه
Euler-type Multiple Integrals as Linear Forms in Zeta Values
0. In 1978, Apéry showed the irrationality of ζ(3) = ∑∞ n=1 1 n3 by giving the approximants `n = unζ(3) − vn ∈ Qζ(3) + Q, un, dnvn ∈ Z, dn = l.c.m.(1, 2, . . . , n), with the property |`n| → ( √ 2 − 1) < 1/e as n → ∞. A similar approach was put forward to show the irrationality of ζ(2) (which is π/6, hence transcendental thanks to Lindemann) but I will concentrate on the case of ζ(3). A few mon...
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The story exposed in this paper starts in 1978, when R. Apéry [Ap] gave a surprising sequence of exercises demonstrating the irrationality of ζ(2) and ζ(3). (For a nice explanation of Apéry’s discovery we refer to the review [Po].) Although the irrationality of the even zeta values ζ(2), ζ(4), . . . for that moment was a classical result (due to L. Euler and F. Lindemann), Apéry’s proof allows ...
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A general hypergeometric construction of linear forms in (odd) zeta values is presented. The construction allows to recover the records of Rhin and Viola for the irrationality measures of ζ(2) and ζ(3), as well as to explain Rivoal’s recent result on infiniteness of irrational numbers in the set of odd zeta values, and to prove that at least one of the four numbers ζ(5), ζ(7), ζ(9), and ζ(11) i...
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Multiple zeta values (MZVs, also called Euler sums or multiple harmonic series) are nested generalizations of the classical Riemann zeta function evaluated at integer values. The fact that an integral representation of MZVs obeys a shuue product rule allows the possibility of a combi-natorial approach to them. Using this approach we prove a longstanding conjecture of Don Zagier about MZVs with ...
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for any collection of positive integers s1, s2, . . . , sl. By definition, Lis(1) = ζ(s), s ∈ Z, s1 ≥ 2, s2 ≥ 1, . . . , sl ≥ 1. (4.2) Taking, as before for multiple zeta values, Lixs(z) := Lis(z), Li1(z) := 1, (4.3) let us extend action of the map Li : w 7→ Liw(z) by linearity on the graded algebra H (not H, since multi-indices are coded by words in H). Lemma 4.1. Let w ∈ H be an arbitrary non...
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ژورنال
عنوان ژورنال: Functiones et Approximatio Commentarii Mathematici
سال: 2007
ISSN: 0208-6573
DOI: 10.7169/facm/1229619663