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 mzveng [2016/03/06 22:42]Уланский Евгений Алесандрович mzveng [2016/05/12 22:35]Уланский Евгений Алесандрович [Multiple zeta values] Предыдущая версия справа и слева Предыдущая версия 2016/05/12 22:35 Уланский Евгений Алесандрович [Multiple zeta values] 2016/05/08 23:05 Уланский Евгений Алесандрович [Literature] 2016/05/08 23:00 Уланский Евгений Алесандрович [Course programm] 2016/03/06 22:42 Уланский Евгений Алесандрович 2016/03/03 22:54 Уланский Евгений Алесандрович [Multiple zeta values] 2016/01/06 12:03 Уланский Евгений Алесандрович создано Следующая версия Предыдущая версия 2016/05/12 22:35 Уланский Евгений Алесандрович [Multiple zeta values] 2016/05/08 23:05 Уланский Евгений Алесандрович [Literature] 2016/05/08 23:00 Уланский Евгений Алесандрович [Course programm] 2016/03/06 22:42 Уланский Евгений Алесандрович 2016/03/03 22:54 Уланский Евгений Алесандрович [Multiple zeta values] 2016/01/06 12:03 Уланский Евгений Алесандрович создано Строка 1: Строка 1: ======Multiple zeta values====== ======Multiple zeta values====== - Special course given **in English** by [[ulanskiy|E. A. Ulanskii]], one term, for students of 2-5 years. **On Wednesdays at 18:30 in 14-14 auditorium**. + Special course given **in English** by [[ulanskiy|E. A. Ulanskii]], one term, for students of 2-5 years. ​ ======Abstract====== ======Abstract====== Строка 9: Строка 9: ======Course programm====== ======Course programm====== - 1) Zeta values. ​Basel problem and its solution by Leonard Euler. + 1) Zeta values. ​Multiple polylogarithms. Differential equations. - 2) Roger Apery’s theorem on irrationality ​of ζ(3). + 2)      ​Integral representation and analitic continuation. General integral ​of hypergeometric type. - 3) Tanguy Rivoal and Keith Ball’s theorem on irrationality of  ​ζ(2n+1) for infinitely many n. + 3) Formulae for ζ(2n). - 4) Wadim Zudilin’s theorem on irrationality ​of at least one of four numbers ​ ζ(5),​ζ(7),​ζ(9),​ζ(11). + 4) Closed formulae for values ​of dilogarithm and trilogarithm. - 5) Closed formulae for zeta values. + 5) Periodic ​zeta values. - 6) Multiple zeta values (MZV) and generalized polylogarithms. Weight and length. Classical polylogarithms. Euler formulae ​for MZV including ​ζ(2,​1)=ζ(3). + 6) Duality for MZVs. Euler formula ​for MZVs. Special case ζ(2,​1)=ζ(3). - 7) Closed formulae ​for MZV and some values ​of generalized polylogarithms. + 7) Standard relations ​for MZVs. Proof of shuffle relations. - 8) Standard ​relations ​for MZV. + 8)      Proof of stuffle ​relations. - 9) Michael Hoffman relations and their connection with standard relations. + 9) Michael Hoffman relations and their connection with standard relations. Proof. - 10) Integral representations ​for MZV and generalized polylogarithms. + 10) Sum formula ​for MZVs. Proof of Okuda and Ueno. - 11) Sum relation for MZV. Duality for MZV. + 11)     Igarashi'​s proof of sum formula. - 12) Yasuo Ohno relations for MZV and their connection with sum formula and duality. + 12) Yasuo Ohno relations for MZVs and their connection with sum formula and duality. Scetch of proof à la Igarashi. - 13) Transformations ​-z/1-z and 1-z for generalized polylogarithms. + 13) Transformation ​-z/1-z for generalized polylogarithms. - 14) Linear independence of generalized polylogarithms. Algebraic independence of classical polylogarithms. + 14) Transformation 1-z for generalized polylogarithms. - + - 15) Colored generalized polylogarithms. Connection between different integral representations of generalized polylogarithms. + - + - 16) Identities ​for integrals of hypergeometric type. Consequences for generalized polylogarithms. + - + - 17) Linear spaces generated by values of generalized polylogarithms ​of fixed weight. + Строка 51: Строка 45: - Hommfan M. E. Multiple harmonic series. Pacific Journal of Mathematics. 152. No 2. 1992. 275-290. - Hommfan M. E. Multiple harmonic series. Pacific Journal of Mathematics. 152. No 2. 1992. 275-290. - Hommfan M. E. The Algebra of Multiple harmonic series. Journal of Algebra. 194. No 2. 1997. 477-495. - Hommfan M. E. The Algebra of Multiple harmonic series. Journal of Algebra. 194. No 2. 1997. 477-495. + - Igarashi M. On generalizations of the sum formula for multiple zeta values. [[http://​arxiv.org/​abs/​1110.4875|arXiv/​1110.4875]],​ 2011. - Landen J. A New Method of Computing the Sums of Certain Series. Philosophical Transactions of the Royal Society of London. 51. 1759. 553-565. - Landen J. A New Method of Computing the Sums of Certain Series. Philosophical Transactions of the Royal Society of London. 51. 1759. 553-565. - Landen J. Mathematical memoirs respecting a variety of subjects: with an appendix containing tables of theorems for the calculation of fluent. Vol. 1. 1780. London: J. Nourse. - Landen J. Mathematical memoirs respecting a variety of subjects: with an appendix containing tables of theorems for the calculation of fluent. Vol. 1. 1780. London: J. Nourse. - Ohno Y. A generalization of the duality and sum formulas on the multiple zeta values. Journal of Number Theory. 74. No. 1. 1999. 39-43. - Ohno Y. A generalization of the duality and sum formulas on the multiple zeta values. Journal of Number Theory. 74. No. 1. 1999. 39-43. + - Okuda J-i., Ueno K. Relations for Multiple Zeta Values and Mellin Transforms of Multiple Polylogarithms.,​ Publ. Res. Inst. Math. Sci. 40 (2004), no. 2, 537-564. - Zagier D. Values of zeta functions and their applications. First European Congress of Mathematics. Birkhauser. Boston. II. 1994. 497-512. - Zagier D. Values of zeta functions and their applications. First European Congress of Mathematics. Birkhauser. Boston. II. 1994. 497-512. - - Zlobin S. A Note on Arithmetical Properties of Multiple Zeta Values. ​arXiv:​math.NT/​0601151 v1 9 Jan 2006. http://​arxiv.org/​abs/​math/​0601151 + - Zlobin S. A Note on Arithmetical Properties of Multiple Zeta Values. ​[[http://​arxiv.org/​abs/​math/​0601151|arXiv:​math.NT/​0601151]] v1 9 Jan 2006. - Zudilin W. One of the numbers ζ(5), ζ(7), ζ(9), ζ(11) is irrational. Uspekhi Mat. Nauk. 56. No 4 2001. 149--150. - Zudilin W. One of the numbers ζ(5), ζ(7), ζ(9), ζ(11) is irrational. Uspekhi Mat. Nauk. 56. No 4 2001. 149--150. - Zudilin W. Algebraic relations for multiple zeta values. Russian Math. Surveys 58. No 1. 2003. 1–29. - Zudilin W. Algebraic relations for multiple zeta values. Russian Math. Surveys 58. No 1. 2003. 1–29.
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