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mzveng [2016/03/06 22:42] Уланский Евгений Алесандрович |
mzveng [2016/05/12 22:35] (текущий) Уланский Евгений Алесандрович [Multiple zeta values] |
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======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====== | ||
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======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. | + | |
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- 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. |