Multiple zeta functions and polylogarithms over global function fields
Journal de théorie des nombres de Bordeaux, Tome 32 (2020) no. 2, pp. 403-438.

Dans [36], Thakur définit des analogues de la fonction zêta multiple sur les corps de fonctions, ζ d (𝔽 q [T];s 1 ,,s d ) et ζ d (K;s 1 ,,s d ), où K est un corps de fonctions global. Les versions étoilées de ces fonctions ont été étudiées par Masri [28]. Nous prouvons des formules de réduction pour ces fonctions étoilées, nous définissons des analogues des polylogarithmes multiples et nous présentons quelques formules pour des valeurs zêta multiples.

In [36], Thakur defines function field analogs of the classical multiple zeta function, namely, ζ d (𝔽 q [T];s 1 ,,s d ) and ζ d (K;s 1 ,,s d ), where K is a global function field. Star versions of these functions were further studied by Masri [28]. We prove reduction formulas for these star functions, extend the construction to function field analogs of multiple polylogarithms, and exhibit some formulas for multiple zeta values.

Reçu le :
Accepté le :
Publié le :
DOI : 10.5802/jtnb.1128
Classification : 11M41, 11R58, 11T55, 14H05
Mots clés : Function field, multiple zeta function, multiple polylogarithms
Basak, Debmalya 1 ; Degré-Pelletier, Nicolas 2 ; Lalín, Matilde N. 2

1 Indian Institute of Science Education and Research (IISER), Kolkata, Mohanpur, West Bengal 741246, India
2 Université de Montréal, Pavillon André-Aisenstadt, Dépt. de mathématiques et de statistique, CP 6128, succ. Centre-ville Montréal, Québec, H3C 3J7, Canada
@article{JTNB_2020__32_2_403_0,
     author = {Basak, Debmalya and Degr\'e-Pelletier, Nicolas and Lal{\'\i}n, Matilde N.},
     title = {Multiple zeta functions and polylogarithms over global function fields},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {403--438},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {32},
     number = {2},
     year = {2020},
     doi = {10.5802/jtnb.1128},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/jtnb.1128/}
}
TY  - JOUR
AU  - Basak, Debmalya
AU  - Degré-Pelletier, Nicolas
AU  - Lalín, Matilde N.
TI  - Multiple zeta functions and polylogarithms over global function fields
JO  - Journal de théorie des nombres de Bordeaux
PY  - 2020
SP  - 403
EP  - 438
VL  - 32
IS  - 2
PB  - Société Arithmétique de Bordeaux
UR  - http://www.numdam.org/articles/10.5802/jtnb.1128/
DO  - 10.5802/jtnb.1128
LA  - en
ID  - JTNB_2020__32_2_403_0
ER  - 
%0 Journal Article
%A Basak, Debmalya
%A Degré-Pelletier, Nicolas
%A Lalín, Matilde N.
%T Multiple zeta functions and polylogarithms over global function fields
%J Journal de théorie des nombres de Bordeaux
%D 2020
%P 403-438
%V 32
%N 2
%I Société Arithmétique de Bordeaux
%U http://www.numdam.org/articles/10.5802/jtnb.1128/
%R 10.5802/jtnb.1128
%G en
%F JTNB_2020__32_2_403_0
Basak, Debmalya; Degré-Pelletier, Nicolas; Lalín, Matilde N. Multiple zeta functions and polylogarithms over global function fields. Journal de théorie des nombres de Bordeaux, Tome 32 (2020) no. 2, pp. 403-438. doi : 10.5802/jtnb.1128. http://www.numdam.org/articles/10.5802/jtnb.1128/

[1] Anderson, Greg W.; Thakur, Dinesh S. Multizeta values for 𝔽 q [t], their period interpretation, and relations between them, Int. Math. Res. Not. (2009) no. 11, pp. 2038-2055 | MR | Zbl

[2] Aoki, Takashi; Kombu, Yasuhiro; Ohno, Yasuo A generating function for sums of multiple zeta values and its applications, Proc. Am. Math. Soc., Volume 136 (2008) no. 2, pp. 387-395 | DOI | MR | Zbl

[3] Aoki, Takashi; Ohno, Yasuo; Wakabayashi, Noriko On generating functions of multiple zeta values and generalized hypergeometric functions, Manuscr. Math., Volume 134 (2011) no. 1-2, pp. 139-155 | DOI | MR | Zbl

[4] Brown, Francis Multiple zeta values and periods: from moduli spaces to Feynman integrals, Combinatorics and physics (Contemporary Mathematics), Volume 539, American Mathematical Society, 2011, pp. 27-52 | DOI | MR | Zbl

[5] Brown, Francis On the decomposition of motivic multiple zeta values, Galois-Teichmüller theory and arithmetic geometry (Advanced Studies in Pure Mathematics), Volume 63, Mathematical Society of Japan, 2012, pp. 31-58 | DOI | MR | Zbl

[6] Chang, Chieh-Yu; Mishiba, Yoshinori On finite Carlitz multiple polylogarithms, J. Théor. Nombres Bordeaux, Volume 29 (2017) no. 3, pp. 1049-1058 | DOI | MR | Zbl

[7] Chang, Chieh-Yu; Mishiba, Yoshinori On multiple polylogarithms in characteristic p: v-adic vanishing versus -adic Eulerianness, Int. Math. Res. Not. (2019) no. 3, pp. 923-947 | DOI | MR | Zbl

[8] Chen, Kwang-Wu; Chung, Chan-Liang Sum relations of multiple zeta star values with even arguments, Mediterr. J. Math., Volume 14 (2017) no. 3, 110, 13 pages | DOI | MR | Zbl

[9] Deligne, Pierre Le groupe fondamental de la droite projective moins trois points, Galois groups over Q (Berkeley, CA, 1987) (Mathematical Sciences Research Institute Publications), Volume 16, Springer, 1989, pp. 79-297 | DOI | MR | Zbl

[10] Drinfeld, Vladimir G. On quasitriangular quasi-Hopf algebras and on a group that is closely connected with Gal (Q ¯/Q), Algebra Anal., Volume 2 (1990) no. 4, pp. 149-181 | MR

[11] Euler, Leonhard Meditationes circa singulare serierum genus, Novi Comm. Acad. Sci. Petropol., Volume 20 (1775), pp. 140-186

[12] Goncharov, Alexander B.; Manin, Yuri I. Multiple ζ-motives and moduli spaces ¯ 0,n , Compos. Math., Volume 140 (2004) no. 1, pp. 1-14 | DOI | MR

[13] Hessami Pilehrood, Khodabakhsh; Hessami Pilehrood, Tatiana On q-analogues of two-one formulas for multiple harmonic sums and multiple zeta star values, Monatsh. Math., Volume 176 (2015) no. 2, pp. 275-291 | DOI | MR | Zbl

[14] Hessami Pilehrood, Khodabakhsh; Hessami Pilehrood, Tatiana; Tauraso, Roberto Multiple harmonic sums and multiple harmonic star sums are (nearly) never integers, Integers, Volume 17 (2017), A10, 12 pages | MR | Zbl

[15] Hessami Pilehrood, Khodabakhsh; Hessami Pilehrood, Tatiana; Zhao, Jianqiang On q-analogs of some families of multiple harmonic sums and multiple zeta star value identities, Commun. Number Theory Phys., Volume 10 (2016) no. 4, pp. 805-832 | DOI | MR | Zbl

[16] Hoffman, Michael E. Multiple harmonic series, Pac. J. Math., Volume 152 (1992) no. 2, pp. 275-290 | DOI | MR | Zbl

[17] Hoffman, Michael E. The algebra of multiple harmonic series, J. Algebra, Volume 194 (1997) no. 2, pp. 477-495 | DOI | MR | Zbl

[18] Ihara, Kentaro; Kajikawa, Jun; Ohno, Yasuo; Okuda, Jun-Ichi Multiple zeta values vs. multiple zeta-star values, J. Algebra, Volume 332 (2011), pp. 187-208 | DOI | MR | Zbl

[19] Kaneko, Masanobu; Ohno, Yasuo On a kind of duality of multiple zeta-star values, Int. J. Number Theory, Volume 6 (2010) no. 8, pp. 1927-1932 | DOI | MR | Zbl

[20] Kondo, Hiroki; Saito, Shingo; Tanaka, Tatsushi The Bowman-Bradley theorem for multiple zeta-star values, J. Number Theory, Volume 132 (2012) no. 9, pp. 1984-2002 | DOI | MR | Zbl

[21] Kontsevich, Maxim Vassiliev’s knot invariants, I. M. Gel’fand Seminar (Advances in Soviet Mathematics), Volume 16, American Mathematical Society, 1993, pp. 137-150 | DOI | MR | Zbl

[22] Kontsevich, Maxim Operads and motives in deformation quantization, Lett. Math. Phys., Volume 48 (1999) no. 1, pp. 35-72 Moshé Flato (1937–1998) | DOI | MR | Zbl

[23] Kontsevich, Maxim; Zagier, Don Periods, Mathematics unlimited—2001 and beyond, Springer, 2001, pp. 771-808 | DOI | MR | Zbl

[24] Li, Zhonghua; Qin, Chen Stuffle product formulas of multiple zeta values, Taiwanese J. Math., Volume 22 (2018) no. 3, pp. 529-543 | DOI | MR | Zbl

[25] Linebarger, Erin; Zhao, Jianqiang A family of multiple harmonic sum and multiple zeta star value identities, Mathematika, Volume 61 (2015) no. 1, pp. 63-71 | DOI | MR | Zbl

[26] Machide, Tomoya Identity involving symmetric sums of regularized multiple zeta-star values, Mosc. J. Comb. Number Theory, Volume 8 (2019) no. 2, pp. 125-136 | DOI | MR | Zbl

[27] Manin, Yuri I. Iterated integrals of modular forms and noncommutative modular symbols, Algebraic geometry and number theory (Progress in Mathematics), Volume 253, Birkhäuser, 2006, pp. 565-597 | DOI | MR | Zbl

[28] Masri, Riad Multiple zeta values over global function fields, Multiple Dirichlet series, automorphic forms, and analytic number theory (Proceedings of Symposia in Pure Mathematics), Volume 75, American Mathematical Society, 2006, pp. 157-175 | DOI | MR | Zbl

[29] Muneta, Shuichi On some explicit evaluations of multiple zeta-star values, J. Number Theory, Volume 128 (2008) no. 9, pp. 2538-2548 | DOI | MR | Zbl

[30] Ohno, Yasuo; Okuda, Jun-Ichi On the sum formula for the q-analogue of non-strict multiple zeta values, Proc. Am. Math. Soc., Volume 135 (2007) no. 10, pp. 3029-3037 | DOI | MR | Zbl

[31] Ohno, Yasuo; Wakabayashi, Noriko Cyclic sum of multiple zeta values, Acta Arith., Volume 123 (2006) no. 3, pp. 289-295 | DOI | MR | Zbl

[32] Ohno, Yasuo; Zudilin, Wadim Zeta stars, Commun. Number Theory Phys., Volume 2 (2008) no. 2, pp. 325-347 | DOI | MR | Zbl

[33] Rosen, Michael Number theory in function fields, Graduate Texts in Mathematics, 210, Springer, 2002, xii+358 pages | DOI | MR | Zbl

[34] Schmidt, Friedrich Karl Analytische Zahlentheorie in Körpern der Charakteristik p, Math. Z., Volume 33 (1931) no. 1, pp. 1-32 | DOI | MR | Zbl

[35] Tasaka, Koji; Yamamoto, Shuji On some multiple zeta-star values of one-two-three indices, Int. J. Number Theory, Volume 9 (2013) no. 5, pp. 1171-1184 | DOI | MR | Zbl

[36] Thakur, Dinesh S. Function field arithmetic, World Scientific, 2004, xvi+388 pages | DOI | MR | Zbl

[37] Thakur, Dinesh S. Relations between multizeta values for 𝔽 q [t], Int. Math. Res. Not. (2009) no. 12, pp. 2318-2346 | DOI | MR

[38] Thakur, Dinesh S. Shuffle relations for function field multizeta values, Int. Math. Res. Not. (2010) no. 11, pp. 1973-1980 | DOI | MR

[39] Thakur, Dinesh S. Multizeta values for function fields: a survey, J. Théor. Nombres Bordeaux, Volume 29 (2017) no. 3, pp. 997-1023 | MR

[40] Xu, Ce Identities for the multiple zeta (star) values, Results Math., Volume 73 (2018) no. 1, 3, 22 pages | DOI | MR

[41] Yamamoto, Shuji Explicit evaluation of certain sums of multiple zeta-star values, Funct. Approximatio, Comment. Math., Volume 49 (2013) no. 2, pp. 283-289 | DOI | MR

[42] Yamazaki, Chika On the duality for multiple zeta-star values of height one, Kyushu J. Math., Volume 64 (2010) no. 1, pp. 145-152 | DOI | MR

[43] Zagier, Don Values of zeta functions and their applications, First European Congress of Mathematics, Vol. II (Paris, 1992) (Progress in Mathematics), Volume 120, Birkhäuser, 1994, pp. 497-512 | MR

[44] Zagier, Don Evaluation of the multiple zeta values ζ(2,...,2,3,2,...,2), Ann. Math., Volume 175 (2012) no. 2, pp. 977-1000 | DOI | MR

[45] Zhao, Jianqiang Identity families of multiple harmonic sums and multiple zeta star values, J. Math. Soc. Japan, Volume 68 (2016) no. 4, pp. 1669-1694 | DOI | MR

Cité par Sources :