## Number Theory Research in Turku

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.

In our group we have also studied and study the theory of automorphic functions, on which for instance advanced analytical machinery is applied to classical number theoretical problems. The topic is of high interest in current analytic number theory with applications to a wide range of important questions in both classical and more recent number theory.

Nowadays the analytic number theory research in Turku is concentrated on multiplicative number theory, including theory of multiplicative functions, sieve methods and their applications, as well as on connections to additive combinatorics.

Algebraic number theory has also been researched in Turku but these days algebraic number theory research concentrates on applying number theory
to coding theoretical problems. Recently the coding theory
group has applied class field theory from algebraic number theory, ergodic methods from analytic number theory and results from Diophantine approximation to problems in information
theory.

## Members

#### Professors Emeriti

- Tauno Metsänkylä

publications

#### Docents

- Reijo Ernvall, Senior lecturer, HAMK University of Applied Sciences

#### Lecturers

#### Researchers

#### Doctoral students

- Jori Merikoski
- Juho Salmensuu
- Joni Teräväinen

#### Some theses of former members of the group, defended in Turku

- Anne-Maria Ernvall-Hytönen (Åbo Akademi)
- Henri Virtanen (MELA)