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

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.

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.

Received:
Accepted:
Published online:
DOI: 10.5802/jtnb.1128
Classification: 11M41,  11R58,  11T55,  14H05
Keywords: Function field, multiple zeta function, multiple polylogarithms
Debmalya Basak 1; Nicolas Degré-Pelletier 2; Matilde N. Lalín 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
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
@article{JTNB_2020__32_2_403_0,
     author = {Debmalya Basak and Nicolas Degr\'e-Pelletier and Matilde N. Lal{\'\i}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 = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1128/}
}
TY  - JOUR
AU  - Debmalya Basak
AU  - Nicolas Degré-Pelletier
AU  - Matilde N. Lalín
TI  - Multiple zeta functions and polylogarithms over global function fields
JO  - Journal de théorie des nombres de Bordeaux
PY  - 2020
DA  - 2020///
SP  - 403
EP  - 438
VL  - 32
IS  - 2
PB  - Société Arithmétique de Bordeaux
UR  - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1128/
UR  - https://doi.org/10.5802/jtnb.1128
DO  - 10.5802/jtnb.1128
LA  - en
ID  - JTNB_2020__32_2_403_0
ER  - 
%0 Journal Article
%A Debmalya Basak
%A Nicolas Degré-Pelletier
%A Matilde N. Lalín
%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 https://doi.org/10.5802/jtnb.1128
%R 10.5802/jtnb.1128
%G en
%F JTNB_2020__32_2_403_0
Debmalya Basak; Nicolas Degré-Pelletier; Matilde N. Lalín. Multiple zeta functions and polylogarithms over global function fields. Journal de théorie des nombres de Bordeaux, Volume 32 (2020) no. 2, pp. 403-438. doi : 10.5802/jtnb.1128. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1128/

[1] Greg W. Anderson; Dinesh S. Thakur 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] Takashi Aoki; Yasuhiro Kombu; Yasuo Ohno 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] Takashi Aoki; Yasuo Ohno; Noriko Wakabayashi 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] Francis Brown 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] Francis Brown 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] Chieh-Yu Chang; Yoshinori Mishiba On finite Carlitz multiple polylogarithms, J. Théor. Nombres Bordeaux, Volume 29 (2017) no. 3, pp. 1049-1058 | DOI | MR | Zbl

[7] Chieh-Yu Chang; Yoshinori Mishiba 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] Kwang-Wu Chen; Chan-Liang Chung 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] Pierre Deligne 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] Vladimir G. Drinfeld 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] Leonhard Euler Meditationes circa singulare serierum genus, Novi Comm. Acad. Sci. Petropol., Volume 20 (1775), pp. 140-186

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

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

[15] Khodabakhsh Hessami Pilehrood; Tatiana Hessami Pilehrood; Jianqiang Zhao 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] Michael E. Hoffman Multiple harmonic series, Pac. J. Math., Volume 152 (1992) no. 2, pp. 275-290 | DOI | MR | Zbl

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

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

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

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

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

[25] Erin Linebarger; Jianqiang Zhao 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] Tomoya Machide 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] Yuri I. Manin 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] Riad Masri 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] Shuichi Muneta On some explicit evaluations of multiple zeta-star values, J. Number Theory, Volume 128 (2008) no. 9, pp. 2538-2548 | DOI | MR | Zbl

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

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

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

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

[35] Koji Tasaka; Shuji Yamamoto 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] Dinesh S. Thakur Function field arithmetic, World Scientific, 2004, xvi+388 pages | DOI | MR | Zbl

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

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

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

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

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

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

[43] Don Zagier 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 | DOI | MR | Zbl

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

[45] Jianqiang Zhao 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 | Zbl

Cited by Sources: