# Number theory group

The number theoretic research at the university of Turku was initiated by late professor Kustaa Inkeri (1908-1997). At the moment the research group is lead by  emeritus professors Matti Jutila and Tauno Metsänkylä and the research is done mostly in algebraic and analytical number theory.

## Algebraic Number Theory

Many questions initially related to integers can be extended to larger domains, namely, to  algebraic number fields. Particularly interesting are such fundamental concepts as the unit group, the discriminant and the class number, which all depend on the arithmetical structure of the field. Deep algebraic methods are used in the study of these concepts.
Algebraic number theory is by no means based solely on algebra. For example, L-functions, which carry valuable information about many invariants associated to number fields, create a strong link to analysis. In addition, the theory of elliptic curves has proved to be immensely useful in many difficult problems concerning algebraic number fields.
The research of our group is concentrated on theoretical and computational questions on Abelian fields, which are Galois extensions of the field of rationals whose Galois group is commutative. The arithmetic of such fields is closely related to p-adic L-functions. We have investigated many questions related to these functions, such as the location of zeros. Moreover, we have worked on Iwasawa theory, whose roots are in certain infinite Abelian extensions, the canonical example of which is the extension generated by all roots of unity of order pn, where p is a fixed prime.

## Analytic Number Theory

In the analytic number theory complex analysis is applied to number theoretical problems. The most important tool is, no doubt, the  Riemann zeta-function z(s), which is an example of L-functions. The validity of the  Riemann Hypothesis  on the location of the zeros of the zeta-function has since Wiles' proof of the Fermat conjecture been the most famous open mathematical question. It has an immediate connection to the distribution of primes; an answer in the affirmative would imply that primes are distributed, in a sense, as evenly as possible among all natural numbers. Our group in Turku have studied many questions related to L-functions: distribution of their zeros, mean value problems, and their connections with so-called  divisor problems, which deal with statistics of the number of divisors of a integer.
Another significant class of tools studied in Turku is that of character and exponential sums, which play an important role both in certain classical problems (such as the Goldbach conjecture and Waring's problem) and in the theory of L-functions.
The latest arrival in research is the spectral theory of automorphic functions, which is based on functional analysis ( i.e. generalized Fourier analysis). The theory has shed light on, for example, the theory of zeta-function.

## Activities

An international conference ( Turku Symposium on Number Theory in Memory of Kustaa Inkeri ), dedicated to his memory, was organized in Turku from May 31 to June 4, 1999. Its proceedings have appeared as a special volume published by Walter De Gruyter & Co. (ed. Matti Jutila and Tauno Metsänkylä).

Number theory days in Finland which gather together Finnish researchers in Number theory was organized in Turku at 2005.

A conference on Algorithmic Number Theory was held in Turku on May 8?11, 2007.