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Number theory and zeta functions

- Speaker：Masataka Chida（The Hakubi Center）
- Date：19th October 2010 (Tuesday), 16:00-
- Venue：The Hakubi Center (iCeMS West Wing 2F, Seminar Room)
- Presentation Language：Japanese

Upon hearing the word mathematics, many people conjure up images of extremely complex and difficult questions. In the area of number theory, we often consider simple problems, like Fermat’s last theorem. One old question in number theory is the distribution of primes. By definition, a prime number has no divisor except for 1 and itself, but it is not easy to understand the distribution of the primes within natural numbers. For this problem, Riemann defined an important function, the so-called the Riemann zeta function, having noticed that the behavior of values of his zeta function is closely related to the distribution of the primes in explained in the 19th century.

In his research, Riemann reached an important conjecture, the so-called Riemann hypothesis. After Riemann, Hadamard and de la Vallee Poussin solved this problem independently of each other using a property of the Riemann zeta function. Their result is called the prime number theorem. However, the Riemann hypothesis remains unsolved to this day, and it seems that we are very far from solving the problem. The zeta function has been generalized in various ways, and therefore zeta functions are still an important research subject in number theory. The automorphic L-function, which is my research subject, is a kind of zeta function, which closely related to the BSD conjecture, and is one of the Millennium Prize Problems of the Clay Mathematical institute. In this seminar, I will explain the zeta function and its relations with my research using some concrete examples.