Koło Naukowe Matematyków U¦ PL SMS

XV SMS' Session
Number Theory

Szczyrk, 12th - 14th March 2004

Because of holidays and other seasons we couldn't book the resort immediately after the exam session, but at last we succeeded. We quickly came up with a topic and we could say farewell to winter with the Queen of Sciences.
For some time now everybody has been having activities on Fridays, so the first day of session begins around 3 o'clock. This time Rafał showed up the first, took the keys and began making salads. Participants (among whom two lovely Anias from the first year gave us big pleasure in showing up) helped him as they were arriving, went shopping and the dinner was ready on time.

After dinner, instead of thanks, our President started talking. For over 2 hours all present listened about arithmetic functions, Dirichlet convolutions and summation formulas and examined the proofs of a few formulas describing the behaviour of certain sums taken over prime numbers, all to see (elementary, to be honest) apparatus needed to prove the Dirichlet Arithmetic Progress Theorem. Participants kept arriving even in the night.

Saturday morning, after spontaneous waking up and equally spontaneous breakfast, was devoted to winter sports: in this case, mostly sledging. It also turned out that mathematicians can also make snowmans and, while standing in knee-deep snow, eat chocolate and solve Oxford riddles. We should also mention that, according to a two-sessions-old tradition, Michał Lewicki went hiking even before breakfast - of course, everyone can do so AFTER breakfast. Our tutor also arrived and, taking our example, went skiing.

For a change, we had spaghetti for lunch. And after lunch what we all like the most: chips, peanuts, juices and talks.

Piotrek Sulich began with a proof of an interesting quality of the Euler "phi" function. He also tried to convince us that his method may be used for a little more general result, and he managed to convince some of us.
After him Marysia Kania told us about the Bernoulli numbers, their long history, many applications and showing up in practically everything that a mathematician can think about.
Radek Wieczorek amazed the participants by presenting a probabilistic approach to number-theoretic problems (younger participants had an occassion to see some probability theory in action). His key theorem was something we could name The Law of Big Prime Numbers with its astonishing consequences, derived from the application of it to estimate average number of prime divisors of natural numbers smaller than a given number.
The best was left for the end: Ikehara theorem and a simple corollary - Prime Numbers Theorem, presented by Tomek Kochanek. Some notions and facts were not elementary, but - broadly speaking - the Tauber Theorems are very powerful.
Dinner was pretty early (for us). And after it the participants divided into groups and social activities lasted until the morning.

The Sunday talk of Rafał was accompanied with farewells of those having to depart earlier. Those few who remained were surprised that instead of some technical results, which could be expected from the abstracts received, the talker first introduced the notion of Dirichlet characters and the orthogonality relation between them, only to spend most of the time on digressions concerning the history of Number Theory and ending with the idea of the proof of the Dirichlet theorem.

As usual, session ended with cleaning up and lunch, which again was beetroot soup. After lunch we had some time to take photos with snowy summits in the background and for purely British in spirit 5'o clock tea.

last update: 15.04.2011


Students' Mathematical Society of the University of Silesia
(Koło Naukowe Matematyków Uniwersytetu ¦l±skiego)
40-007 Katowice, ul. Bankowa 14 (room 524)
tel. (032) 359-20-96, email: knm@knm.katowice.pl