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The multiple zeta value data mine. (English) Zbl 1221.11183

The authors provide a data mine of proven results for Multiple Zeta Values (MZVs) and Euler sums. They investigate the Euler sums to weight \(w=12\) completely, deriving basis-representations for all individual values in an explicit analytic calculation. For the MZVs the same analysis is being performed up to \(w=22\). To \(w=24\) they checked the conjectured size of the basis using modular arithmetic. Under the further conjecture that the basis elements can be chosen out of MZVs of depth \(d\leq \frac{w}{3}\) they confirm the conjecture up to \(w=26\).
The following runs at limited depth, using modular arithmetic keeping the highest weight terms only, were performed: \(d=7, w=27;d=6,w=28;d=7,w=29;d=6,w=30\). For the Euler sums complete results were obtained for \(d\leq 3,w=29;d\leq 4,w=22;d\leq 5,w=17\) and for \(d\leq 3,w=51;d\leq 4,w=30;d\leq 5,w=21;d\leq 6,w=17\) using modular arithmetic neglecting products of lower weight.

MSC:

11M32 Multiple Dirichlet series and zeta functions and multizeta values
11Y35 Analytic computations

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References:

[1] Euler, L., Meditationes circa singulare serium genus, Novi Comm. Acad. Sci. Petropol.. (Opera Omnia Ser. I, vol. 15 (1927), B.G. Teubner: B.G. Teubner Berlin), 20, 217-267 (1775), reprinted
[2] Zagier, D., Values of zeta functions and their applications, (First European Congress of Mathematics, vol. II. First European Congress of Mathematics, vol. II, Paris, 1992. First European Congress of Mathematics, vol. II. First European Congress of Mathematics, vol. II, Paris, 1992, Progr. Math., vol. 120 (1994), Birkhäuser: Birkhäuser Basel-Boston), 497-512 · Zbl 0822.11001
[3] For an extended list of references, see: M.E. Hoffman’s page
[4] Nielsen, N., Handbuch der Theorie der Gammafunktion (1906), Teubner: Teubner Leipzig, Reprint of · JFM 37.0450.01
[6] Cartier, P., Fonctions polylogarithmes, nombres polyzêtas et groupes pro-unipotents, Sém. Bourbaki, Mars 2001, 53e année. Sém. Bourbaki, Mars 2001, 53e année, Asterisque, 282, 885, 137-173 (2002) · Zbl 1085.11042
[7] Zudilin, V. V., Algebraic relations for multiple zeta values, Uspekhi Mat. Nauk. Uspekhi Mat. Nauk, Math. Surveys, 58, 1, 1-29 (2003) · Zbl 1171.11323
[8] Devoto, A.; Duke, D. W., Table of integrals and formulae for Feynman diagram calculations, Riv. Nuovo Cim., 7, 6, 1-39 (1984)
[9] Gonzalez-Arroyo, A.; Lopez, C.; Yndurain, F. J., Second order contributions to the structure functions in deep inelastic scattering. 1. Theoretical calculations, Nucl. Phys. B, 153, 161-186 (1979)
[10] Vermaseren, J. A.M., Harmonic sums, Mellin transforms and integrals, Int. J. Modern Phys. A, 14, 2037-2076 (1999) · Zbl 0939.65032
[11] Blümlein, J.; Kurth, S., Harmonic sums and Mellin transforms up to two-loop order, Phys. Rev. D, 60, 014018, 31 (1999)
[12] Broadhurst, D. J., On the enumeration of irreducible \(k\)-fold Euler sums and their roles in knot theory and field theory
[13] Broadhurst, D. J.; Kreimer, D., Association of multiple zeta values with positive knots via Feynman diagrams up to 9 loops, Phys. Lett. B, 393, 403-412 (1997) · Zbl 0946.81028
[14] Deligne, P.; Goncharov, A. B., Groupes fondamentaux motiviques de Tate mixtes, Ann. Sci. Ecole Norm. Sup., Série IV, 38, 1, 1-56 (2005) · Zbl 1084.14024
[15] Bailey, D. H.; Broadhurst, D. J., Parallel integer relation detection: Techniques and applications, Math. Comp., 70, 236, 1719-1736 (2001), (electronic) · Zbl 1037.11090
[16] Lenstra, A. K.; Lenstra, H. W.; Lovasz, L., Factoring polynomials with rational coefficients, Math. Ann., 261, 515-534 (1982) · Zbl 0488.12001
[17] Chen, K. T., Iterated integrals of differential forms and loop space homology, Ann. of Math., 97, 217-246 (1973) · Zbl 0227.58003
[18] Remiddi, E.; Vermaseren, J. A.M., Harmonic polylogarithms, Int. J. Mod. Phys. A, 15, 725-754 (2000) · Zbl 0951.33003
[19] Borwein, J. M.; Bradley, D. M.; Broadhurst, D. J., Evaluation of \(k\)-fold Euler/Zagier sums: A compendium of results for arbitrary \(k\), Electron. J. Combin., 4, 2 (1997), #R5 · Zbl 0884.40004
[21] Vermaseren, J. A.M., New features of FORM · Zbl 1344.65050
[22] Tentyukov, M.; Vermaseren, J. A.M., The multithreaded version of FORM
[23] Vermaseren, J. A.M.; Vogt, A.; Moch, S., The third-order QCD corrections to deep-inelastic scattering by photon exchange, Nucl. Phys. B, 724, 3-182 (2005) · Zbl 1178.81286
[24] Blümlein, J.; Kauers, M.; Klein, S.; Schneider, C., Determining the closed forms of the \(O(a_s^3)\) anomalous dimensions and Wilson coefficients from Mellin moments by means of computer algebra, Comput. Phys. Commun., 180, 2143-2165 (2009) · Zbl 1197.81037
[25] Blümlein, J.; Klein, S., Structural relations between harmonic sums up to \(w = 6, 5\) p · Zbl 1197.81036
[26] Blümlein, J., Structural relations of harmonic sums and Mellin transforms at weight \(w = 6\), Clay Mathematical Institute Proceedings, in press · Zbl 1218.81103
[27] Broadhurst, D. J.; Gracey, J. A.; Kreimer, D., Beyond the triangle and uniqueness relations: Non-zeta counterterms at large N from positive knots, Z. Phys. C, 75, 559-574 (1997)
[28] Broadhurst, D. J., Massive 3-loop Feynman diagrams reducible to SC* primitives of algebras of the sixth root of unity, Eur. Phys. J. C, 8, 311-333 (1999)
[29] Andre, Y., Ambiguity theory, old and new, Bollettino U.M.I. (8) I (2008), in press, and talk at the Motives, Quantum Field Theory, and Pseudodifferential Operators, Boston University, June 2-13, 2008
[30] Brown, F., The massless higher-loop two-point function, Comm. Math. Phys., 287, 925-958 (2009) · Zbl 1196.81130
[31] Kontsevich, M.; Zagier, D., Periods, (Mathematics Unlimited-2001 and Beyond (2001), Springer: Springer Berlin), 771-808 · Zbl 1039.11002
[32] Bogner, C.; Weinzierl, S., J. Math. Phys., 50, 042302 (2009)
[33] Lewin, L., Polylogarithms and Associated Functions (1981), North-Holland: North-Holland New York · Zbl 0465.33001
[34] Kölbig, K. S., Nielsen’s generalized polylogarithms, SIAM J. Math. Anal., 17, 1232-1258 (1986) · Zbl 0606.33013
[35] Moch, S.; Uwer, P.; Weinzierl, S., Nested sums, expansion of transcendental functions and multi-scale multi-loop integrals, J. Math. Phys., 43, 3363-3386 (2002) · Zbl 1060.33007
[36] Goncharov, A. B., Multiple polylogarithms, cyclotomy and modular complexes, Math. Res. Lett., 5, 497-516 (1998) · Zbl 0961.11040
[37] Blümlein, J., Algebraic relations between harmonic sums and associated quantities, Comput. Phys. Commun., 159, 19-54 (2004) · Zbl 1097.11063
[38] Reutenauer, C., Free Algebras (1993), Calendron Press: Calendron Press Oxford
[39] Hoffman, M. E., Algebraic Aspects of Multiple Zeta Values, (Aoki, T.; etal., Zeta Functions, Topology and Quantum Physics. Zeta Functions, Topology and Quantum Physics, Dev. Math., vol. 14 (2005), Springer: Springer New York), 51-74 · Zbl 1170.11324
[40] Hoffman, M. E., Multiple harmonic series, Pacific J. Math., 152, 275-290 (1992) · Zbl 0763.11037
[41] Hasse, H., Ein Summierungsverfahren für die Riemannsche \(ζ\)-Reihe, Math. Z., 32, 458-464 (1930) · JFM 56.0894.03
[42] Sigler, L. E., Fibonacci’s Liber Abaci (2002), Springer: Springer Berlin
[44] Witt, E., Die Unterring der freien Lieschen Ringe, Math. Z., 64, 195-216 (1956) · Zbl 0070.02903
[45] Williams, A.; Shanks, D., Strong primality tests that are not sufficient, Math. Comp., 39, 159, 255-300 (1982) · Zbl 0492.10005
[46] Lucas, E., Théorie des fonctions numériques simplement périodiques, Amer. J. Math., 1, 197-240 (1878)
[47] Borwein, J. M.; Bradley, D. M.; Broadhurst, D. J.; Lisonek, P., Special values of multiple polylogarithms, Trans. Amer. Math. Soc., 353, 907-941 (2001) · Zbl 1002.11093
[48] Borwein, J. M.; Girgensohn, R., Evaluation of triple Euler sums, Electron. J. Combin., 3 (1996), #R23 · Zbl 0884.40005
[49] Markett, C., Triple sums and the Riemann zeta function, J. Number Theory, 48, 113-132 (1994) · Zbl 0810.11047
[50] Espie, M.; Novelli, J.-C.; Racinet, G., Formal computations about multiple zeta values, (Fauvet, F.; Mitschi, C., From Combinatorics to Dynamical Systems. From Combinatorics to Dynamical Systems, Strasbourg, 2002. From Combinatorics to Dynamical Systems. From Combinatorics to Dynamical Systems, Strasbourg, 2002, IMRA Lect. Math. Theor. Phys., vol. 3 (2003), de Gryter: de Gryter Berlin), 1-16, Preprint SFB-478, Univ. Münster Heft 260, 2003 · Zbl 1050.11065
[51] Kaneko, M.; Noro, M.; Tsurumaki, K., On a conjecture for the dimension of the space of the multiple zeta values, Software for Algebraic Geometry IMA, 148, 47-58 (2008) · Zbl 1143.14301
[52] Hoffmann, M. E.; Ohno, Y., Relations of multiple zeta values and their algebraic expressions, J. Algebra, 262, 332-347 (2003) · Zbl 1139.11322
[54] Gastmans, R.; Troost, W., On the evaluation of polylogarithmic integrals, Bull. Belg. Math. Soc. Simon Stevin, 55, 205-219 (1981), KUL-TF-80/10, MR 0647134 (83c:65028) · Zbl 0477.33011
[56] The PARI/GP page
[57] Bailey, D. H.; Borwein, J. M.; Girgensohn, R., Experimental evaluation of Euler sums, Experiment. Math., 3, 17-30 (1994) · Zbl 0810.11076
[59] Bigotte, M.; Jacob, G.; Oussous, N. E.; Petitot, M., Lyndon words and shuffle algebras for generating the coloured multiple zeta values relation tables, Theoret. Comput. Sci., 273, 271-282 (2002) · Zbl 1014.68126
[60] Minh, H. N.; Petitot, M., Lyndon words, polylogarithms and the Riemann \(ζ\) function, Discrete Math., 217, 273-292 (2000) · Zbl 0959.68144
[62] Minh, H. N.; Jacob, G.; Oussuous, N. E.; Petitot, M., Aspects combinatoires des polylogarithmes et des sommes d’Euler-Zagier, J. Électr. Sém. Lothar. Combin., 43 (2000), Art. B43e, 29 pp
[63] Vermaseren, J. A.M. (Nov. 2003)
[65] Vermaseren, J. A.M.
[66] Vermaseren, J. A.M., Tuning form with large calculations, Nucl. Phys. B Proc. Suppl., 116, 343-347 (2003)
[69] Hoffman, M. E., The algebra of multiple harmonic series, J. Algebra, 194, 477-495 (1997) · Zbl 0881.11067
[70] Bowman, D.; Bradley, D. M., The algebra and combinatorics of shuffles and multiple zeta values, J. Combin. Theory Ser. A, 97, 43-61 (2002) · Zbl 1021.11026
[71] Zhao, J., Double shuffle relations of Euler sums
[73] Okuda, J.; Ueno, K., The sum formula of multiple zeta values and connection problem of the formal Knizhnik-Zamolodchikov equation, (Aoki, T.; etal., Zeta Functions, Topology and Quantum Physics. Zeta Functions, Topology and Quantum Physics, Dev. Math., vol. 14 (2005), Springer: Springer New York), 145-170 · Zbl 1170.11327
[75] Le, T. Q.T.; Murakami, J., Kontsevich’s integral for the Homfly polynomial and relations between values of multiple zeta functions, Topology Appl., 62, 193-206 (1995) · Zbl 0839.57007
[76] Ohno, Y., A generalization of the duality and sum formulas on the multiple zeta values, J. Number Theory, 74, 189-209 (1999)
[77] Ohno, Y.; Zagier, D., Multiple zeta values of fixed weight, depth, and height, Indag. Math. (N.S.), 12, 483-487 (2001) · Zbl 1031.11053
[78] Ohno, Y.; Wakabayashi, N., Cyclic sum of multiple zeta values, Acta Arith., 123, 289-295 (2006) · Zbl 1156.11038
[79] Ihara, K.; Kaneko, M.; Zagier, D., Derivation and double shuffle relations for multiple zeta values, Compos. Math., 142, 307-338 (2006), preprint MPIM2004-100 · Zbl 1186.11053
[81] Zhao, J., Linear relations of special values of multiple polylogarithms at roots of unity
[82] Furusho, H., The multiple zeta value algebra and the stable derivation algebra, Publ. RIMS Kyoto Univ., 39, 695-720 (2003) · Zbl 1115.11055
[83] Hardy, G. H.; Wright, E. M., An Introduction to the Theory of Numbers (2002), Calendron Press: Calendron Press Oxford · Zbl 0020.29201
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