Geodesic
Algebraic relations for multiple zeta values
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 58 (2003) no. 1, pp. 1-29 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

The survey is devoted to the multidimensional generalization of the Riemann zeta function as a function of a positive integral argument. 
@article{RM_2003_58_1_a0,
     author = {W. V. Zudilin},
     title = {Algebraic relations for multiple zeta values},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {1--29},
     year = {2003},
     volume = {58},
     number = {1},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/RM_2003_58_1_a0/}
}
TY  - JOUR
AU  - W. V. Zudilin
TI  - Algebraic relations for multiple zeta values
JO  - Trudy Matematicheskogo Instituta imeni V.A. Steklova
PY  - 2003
SP  - 1
EP  - 29
VL  - 58
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/RM_2003_58_1_a0/
LA  - en
ID  - RM_2003_58_1_a0
ER  - 
%0 Journal Article
%A W. V. Zudilin
%T Algebraic relations for multiple zeta values
%J Trudy Matematicheskogo Instituta imeni V.A. Steklova
%D 2003
%P 1-29
%V 58
%N 1
%U http://geodesic.mathdoc.fr/item/RM_2003_58_1_a0/
%G en
%F RM_2003_58_1_a0
W. V. Zudilin. Algebraic relations for multiple zeta values. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 58 (2003) no. 1, pp. 1-29. http://geodesic.mathdoc.fr/item/RM_2003_58_1_a0/

[1] R. Apéry, “Irrationalité de $\zeta(2)$ et $\zeta(3)$”, Astérisque, 61 (1979), 11–13 | MR | Zbl

[2] J. Borwein, D. Bradley, “Empirically determined Apéry-like formulae for $\zeta(4n+3)$”, Experiment. Math., 6:3 (1997), 181–194 | MR | Zbl

[3] J. M. Borwein, D. M. Bradley, D. J. Broadhurst, “Evaluations of $k$-fold Euler/Zagier sums: A compendium of results for arbitrary $k$”, Electron. J. Combin., 4:2 (1997), #R5 | MR | Zbl

[4] J. M. Borwein, D. M. Bradley, D. J. Broadhurst, P. Lisoněk, “Special values of multiple polylogarithms”, Trans. Amer. Math. Soc., 353:3 (2001), 907–941 | DOI | MR | Zbl

[5] L. Euler, “Meditationes circa singulare serierum genus”, Novi Comm. Acad. Sci. Petropol., 20 (1775), 140–186; “Reprinted”, Opera Omnia Ser. I, 15, Teubner, Berlin, 1927, 217–267

[6] A. O. Gelfond, Ischislenie konechnykh raznostei, Nauka, M., 1967 | MR

[7] A. B. Goncharov, “Polylogarithms in arithmetic and geometry”, Proceedings of the International Congress of Mathematicians (ICM'94, Zürich, August 3–11, 1994), I, ed. S. D. Chatterji, Birkhäuser, Basel, 1995, 374–387 | MR | Zbl

[8] A. B. Goncharov, “The double logarithm and Manin's complex for modular curves”, Math. Res. Lett., 4:5 (1997), 617–636 | MR | Zbl

[9] E. Heine, “Über die Reihe ...”, J. Reine Angew. Math., 32 (1846), 210–212 | Zbl

[10] M. E. Hoffman, “Multiple harmonic series”, Pacific J. Math., 152:2 (1992), 275–290 | MR | Zbl

[11] M. E. Hoffman, “The algebra of multiple harmonic series”, J. Algebra, 194:2 (1997), 477–495 | DOI | MR | Zbl

[12] M. E. Hoffman, “Quasi-shuffle products”, J. Algebraic Combin., 11:1 (2000), 49–68 | DOI | MR | Zbl

[13] M. E. Hoffman, Y. Ohno, Relations of multiple zeta values and their algebraic expression, , 2000 arXiv: math/0010140 | MR

[14] K. Ihara, M. Kaneko, Derivation relations and regularized double shuffle relations of multiple zeta values, Preprint, 2000

[15] M. Kaneko, N. Kurokawa, M. Wakayama, A variation of Euler's approach to values of the Riemann zeta function, arXiv: math/0206171

[16] M. Koecher, “Letter to the editor”, Math. Intelligencer, 2:2 (1979/1980), 62–64 | MR

[17] F. Lindemann, “Über die Zahl $\pi$”, Math. Ann., 20 (1882), 213–225 | DOI | MR

[18] H. M. Minh, G. Jacob, M. Petitot, N. E. Oussous, “Aspects combinatoires des polylogarithmes et des sommes d'Euler–Zagier”, Sém. Lothar. Combin., 43 (1999), Art. B43e (electronic) ; http://www.mat.univie.ac.at/~slc/wpapers/s43minh.html | Zbl

[19] H. M. Minh, M. Petitot, “Lyndon words, polylogarithms and the Riemann $\zeta$ function”, Formal Power Series and Algebraic Combinatorics (Vienna, 1997), Discrete Math., 217, no. 1–3, 2000, 273–292 | MR | Zbl

[20] H. M. Minh, M. Petitot, J. van der Hoeven, “Shuffle algebra and polylogarithms”, Formal Power Series and Algebraic Combinatorics (Toronto, ON, June 1998), Discrete Math., 225, no. 1–3, 2000, 217–230 | MR | Zbl

[21] Y. Ohno, “A generalization of the duality and sum formulas on the multiple zeta values”, J. Number Theory, 74:1 (1999), 39–43 | DOI | MR | Zbl

[22] T. Rivoal, “La fonction zêta de Riemann prend une infinité de valeurs irrationnelles aux entiers impairs”, C. R. Acad. Sci. Paris Sér. I Math., 331:4 (2000), 267–270 | MR | Zbl

[23] K.-G. Schlesinger, Some remarks on $q$-deformed multiple polylogarithms, arXiv: math/0111022

[24] V. N. Sorokin, “O lineinoi nezavisimosti znachenii obobschennykh polilogarifmov”, Matem. sb., 192:8 (2001), 139–154 | MR | Zbl

[25] E. A. Ulanskii, “Tozhdestva dlya obobschennykh polilogarifmov”, Matem. zametki, 73:4 (2003), 613–624 | MR | Zbl

[26] D. V. Vasilev, “Nekotorye formuly dlya dzeta-funktsii v tselykh tochkakh”, Vestn. MGU. Ser. 1. Matem., mekh., 1996, no. 1, 81–84

[27] M. Waldschmidt, “Valeurs zêta multiples: une introduction”, J. Théor. Nombres Bordeaux, 12:2 (2000), 581–595 | MR | Zbl

[28] M. Waldschmidt, “Multiple polylogarithms”, Lectures at Institute of Math. Sciences (Chennai, November 2000) | Zbl

[29] E. T. Uitteker, Dzh. N. Vatson, Kurs sovremennogo analiza, t. 2, Fizmatlit, M., 1963

[30] D. Zagier, “Values of zeta functions and their applications”, First European Congress of Mathematics, V. II (Paris, 1992), Progr. Math., 120, ed. A. Joseph et al., Birkhäuser, Boston, 1994, 497–512 | MR | Zbl

[31] J. Zhao, “Analytic continuation of multiple zeta functions”, Proc. Amer. Math. Soc., 128:5 (2000), 1275–1283 | DOI | MR | Zbl

[32] S. A. Zlobin, “Integraly, predstavlyaemye v vide lineinykh form ot obobschennykh polilogarifmov”, Matem. zametki, 71:5 (2002), 782–787 | MR | Zbl

[33] V. V. Zudilin, “Sovershenno uravnoveshennye gipergeometricheskie ryady i kratnye integraly”, UMN, 57:4 (2002), 177–178 | MR | Zbl

[34] V. V. Zudilin, “O diofantovykh zadachakh dlya $q$-dzeta-znachenii”, Matem. zametki, 72:6 (2002), 936–940 | MR | Zbl