Transcendental Number theory: Elimination theory, linear independence and algebraic independence, irrationality of zeta values. (10th Jan - 10th Feb 2011)
Concerning my lectures. I plan to give about 12 one-hour lectures and include the following subjects. 1. Elimination theory and transcendental numbers. 2. Criteria of linear independence. 3. Linear independence of zeta-values (elementary version). 4. Irrationality of \zeta(3) (elementary version). 5. Rational approximations to logarithms of rational numbers. 6. Algebraic independence of \pi and e^\pi. 7. Measure of algebraic independence for almost all numbers. The proof of irrationality of \zeta(3) is new and I hope the simplest one. It follows Apery's proof but uses two time more dense sequence of diophantine approximations. This essentialy simplifies the proof of the recurrence equation, that looks rather simple. Finally the Poincare theorem is used. Due to the very simple situation the version of the Poincare theorem is rather simple too. So I can include it in the lectures. The proof of linear independence for zeta-values mainly follows to Rivoal's theorem. But there is a possibility to exclude all integrals from the proof. After that the proof becomes even simpler. I plan to give a simplified proof of the recent Marcovecchio's result about approximations of log 2 by rational numbers, the only one dimentional complex integrals will be used. I will not discuss zero bounds in connection of \pi and e^\pi and only outline the Philippon's criteria.