# Multiple zeta values

Special course given in English by E. A. Ulanskii, one term, for students of 2-5 years.

# Abstract

This special course covers results from the times of Jacob Bernoulli and Leonard Euler until nowadays. You will find out how one of the most famous classical problems and its magnificent solution have led to appearance of fascinating branch of modern Number theory. You will get to know the proofs of charming theorems and will hear about open problems as modern as with three hundred year history.

# Course programm

1) Zeta values. Multiple polylogarithms. Differential equations.

2) Integral representation and analitic continuation. General integral of hypergeometric type.

3) Formulae for ζ(2n).

4) Closed formulae for values of dilogarithm and trilogarithm.

5) Periodic zeta values.

6) Duality for MZVs. Euler formula for MZVs. Special case ζ(2,1)=ζ(3).

7) Standard relations for MZVs. Proof of shuffle relations.

8) Proof of stuffle relations.

9) Michael Hoffman relations and their connection with standard relations. Proof.

10) Sum formula for MZVs. Proof of Okuda and Ueno.

11) Igarashi's proof of sum formula.

12) Yasuo Ohno relations for MZVs and their connection with sum formula and duality. Scetch of proof à la Igarashi.

13) Transformation -z/1-z for generalized polylogarithms.

14) Transformation 1-z for generalized polylogarithms.

# Literature

1. Granville A. A decomposition of Riemann's zeta-function. Analytic Number Theory. London Mathematical Society Lecture Note Series 247. Y. Motohashi (ред.). Cambridge University Press. 1997. 95-101.
2. Hommfan M. E. Multiple harmonic series. Pacific Journal of Mathematics. 152. No 2. 1992. 275-290.
3. Hommfan M. E. The Algebra of Multiple harmonic series. Journal of Algebra. 194. No 2. 1997. 477-495.
4. Igarashi M. On generalizations of the sum formula for multiple zeta values. arXiv/1110.4875, 2011.
5. Landen J. A New Method of Computing the Sums of Certain Series. Philosophical Transactions of the Royal Society of London. 51. 1759. 553-565.
6. 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.
7. 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.
8. 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.
9. Zagier D. Values of zeta functions and their applications. First European Congress of Mathematics. Birkhauser. Boston. II. 1994. 497-512.
10. Zlobin S. A Note on Arithmetical Properties of Multiple Zeta Values. arXiv:math.NT/0601151 v1 9 Jan 2006.
11. 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.
12. Zudilin W. Algebraic relations for multiple zeta values. Russian Math. Surveys 58. No 1. 2003. 1–29.
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