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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+1for 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.