Geodesic
On multiple integrals represented as a linear form in $1,\zeta(3),\zeta(5),\dots,\zeta(2k-1)$
Fundamentalʹnaâ i prikladnaâ matematika, Tome 11 (2005) no. 6, pp. 143-178 
V. Kh. Salikhov; A. I. Frolovichev. On multiple integrals represented as a linear form in $1,\zeta(3),\zeta(5),\dots,\zeta(2k-1)$. Fundamentalʹnaâ i prikladnaâ matematika, Tome 11 (2005) no. 6, pp. 143-178. http://geodesic.mathdoc.fr/item/FPM_2005_11_6_a13/
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     title = {On multiple integrals represented as a~linear form in $1,\zeta(3),\zeta(5),\dots,\zeta(2k-1)$},
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Voir la notice de l'article provenant de la source Math-Net.Ru

A theorem on the presentability of a multiple integral as a linear form in $1,\zeta(3),\zeta(5),\dots,\zeta(2k-1)$ over $\mathbb Q$ is proved. This theorem refines the results recently obtained by D. Vasiliev, V. Zudilin, and S. Zlobin.

[1] Beitman G., Erdeii A., Vysshie transtsendentnye funktsii, T. 1, Nauka, M., 1973

[2] Vasilev D. V., “Nekotorye formuly dlya znachenii dzeta-funktsii Rimana v tselykh tochkakh”, Vestn. Mosk. un-ta. Ser. 1, Matematika, mekhanika, 1996, no. 1, 81–84

[3] Vasilev D. V., O malykh lineinykh formakh ot znachenii dzeta-funktsii Rimana v nechetnykh tochkakh, Preprint No 1(558), NAN Belarusi, In-t matematiki, Minsk, 2000

[4] Zlobin S. A., “Razlozheniya kratnykh integralov v lineinye formy”, Mat. zametki, 77:5 (2005), 683–706 | MR | Zbl

[5] Beukers F., “A note on the irrationality of $\zeta(2)$ and $\zeta(3)$”, Bull. London Math. Soc., 11 (1979), 268–272 | DOI | MR | Zbl

[6] Zudilin W., “Arithmetics of linear forms involving odd zeta values”, J. Théor. Nombres Bordeaux, 16:1 (2004), 251–291 | MR | Zbl

[7] Zudilin W., “Well-poised hypergeometric transformations of Euler-type multiple-integrals”, J. London Math. Soc. (2), 70 (2004), 215–230 | DOI | MR | Zbl