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