university of tampere:
sis/luo-coms:
research:
cis - the tampere research center for information and systems:
research groups:

- CIRCMI
- DARG
- eHealth
- Logic Group
- MESSI
- NGIS
- »Number Theory Group
- Statistics Group

Number theory is the branch of mathematics concerned with the study of the properties of the integers. It is one of the oldest parts of mathematics, alongside geometry, and has been studied at least since the ancient Mesopotamians and Egyptians.

Problems in number theory are often easy to understand but are solved using sophisticated techniques from different branches of mathematics such as algebra and analysis. Number theory as other fields in mathematics is divided into several classes according to the methods used and the type of questions investigated. See MSC2010 database.

As examples of unsolved conjectures in number theory one can mention the twin-prime problem, the Goldbach conjecture (1742), and the odd perfect number problem. The most important unsolved problem in number theory is probably the Riemann Hypothesis (1859), which states that all non-trivial zeros of the Riemann zeta function have real part equal to one-half.

Gauss, often known as the "prince of mathematics," called mathematics the "queen of the sciences" and considered number theory the "queen of mathematics".

Number theory has various applications in computer science, for ex. in cryptography.

Tampere University number theory group is interested in number theoretic functions and matrices, computer-aided research and mathematics education, and their connection to information technology.

We study basic properties including convolutions, multiplicative functions, periodic functions, Ramanujan–Fourier series and incidence functions on posets. We co-operate with Emil Daniel Schwab (University of Texas at El Paso, USA), V. Sitaramaiah (India) and Laszlo Toth (University of Pecs, Hungary). We have shown among others that the elements of commutative Möbius monoids possess a unique factorization into product of prime elements. We have applied the Cauchy residue theorem to find the order of the usual product of rational arithmetic functions, which was left open by the famous Indian mathematician R. Vaidyanathaswamy in *Trans. Amer. Math. Soc.* (1931). The same applies to the usual product of linear recurrence relations and discrete-time signals. Examples of research papers are

- A unique factorization in commutative Mobius monoids
- An analogue of Ramanujan's sum with respect to regular integers (mod r)
- Arithmetic Incidence Functions: A Study of Factorability
- On the usual product of rational arithmetic functions

Our main topic is GCD related matrices, in particular meet matrices in lattices. Co-operation is made with researchers from P. R. China, Turkey and France. Pauliina Ilmonen, Ismo Korkee and Juha Sillanpää have also had a great impact in our group. Our current interest is in eigenvalues and norms of meet matrices. Examples of research papers can be found in

- On Meet and Join Matrices Associated with Incidence Functions
- On unitary analogs of GCD reciprocal LCM matrices
- A topological approach to divisibility of arithmetical functions and GCD matrices
- On the positive definiteness and eigenvalues of meet and join matrices

Our results on number theoretic functions and matrices have applications in information technology, see e.g. G. Chen, S. Krishnan, T. D. Bui. Matrix-Based Ramanujan-Sums Transforms. IEEE Signal Processing Letters, Vol. 20, No. 10 (2013).

We utilize computer to find solutions to open problem and to generate number-theoretic and related conjectures. We have two projects. One project relies on SURVO computations made by Emeritus Professor Seppo Mustonen (University of Helsinki). He is interested in e.g. the number of certain objects in rectangular grids and properties of regular polygons, and he has found experimentally various interesting results. Our purpose is to prove mathematically the observations and conjectures made by Professor Mustonen. These observations concern, among others, recursions and asymptotic behaviour of the number of gridlines. We have made joint work with Doctors Anne-Maria Ernvall-Hytönen (University of Helsinki) and Kaisa Matomäki (University of Turku) in proving the conjectures. We have shown that one of Mustonen conjectures holds under the Riemann Hypothesis. It is an open problem whether this Mustonen conjecture is equivalent to the Riemann hypothesis. Our results can also be expressed in terms of logic and digital technique. Examples of research papers are

- Formulas for the number of gridlines
- Asymptotics of the number of threshold functions on a two-dimensional rectangular grid
- Survo papers by Mustonen

The other project is related to matrices in number theory. We utilize mathematical software (e.g. SAGE and Mathematica) to find the crucial lattice theoretic structures to explain the behaviour of basic properties, e.g. singularity, of certain LCM type matrices. We have solved, e.g., the Bourque–Ligh conjecture and several conjectures of Hong with the aid of this strategy. It appears that the cubic lattice is essential in the Bourque-Ligh conjecture. We have found the first singular odd number (in the sense of Hong). It is 1020180525, which comes inserting a suitable element in the cubic lattice. These can be found from

- On Smith's determinant
- A lattice-theoretic approach to the Bourque-Ligh conjecture
- Studying the singularity of LCM-type matrices via semilattice structures and their Möbius functions
- On Eigenvalues of Meet and Join Matrices Associated with Incidence Functions

We aim to find connections between school mathematics and university mathematics. We have studied e.g. axioms of perpendicularity and algebraic properties of decimal expressions of rational numbers. We also make empirical and phenomenographic research on mathematics learning with special interest in students’ concept images. Joint work is made with Markku Halmetoja (Mäntän lukio, Finland), Teuvo Laurinolli (Oulun lyseon lukio, Finland), Martti Pesonen (University of East Finland), Jaska Poranen (School of Education, University of Tampere), Ari Virtanen (SIS, University of Tampere). Examples of research papers are

- On direct and inverse proportionality
- Axiomatizing perpendicularity and parallelism
- Different aspects of the monotonicity of a function
- Didactic number theory and group theory for school teachers

A more comprehensive list of publications can be found in Mathscinet.

Pentti Haukkanen (Group leader)

Mika Mattila (Ph. D., Teacher at Tampere University of Technology)

Rauno Soppi (Lic. Phil., Ph. D. Student)

Jori Mäntysalo (M. Sc., interested in Computer-aided Research)

Jorma Merikoski (Emeritus professor, interested in Linear Algebra and Mathematics Education)

Timo Tossavainen (Docent at UTA, Lecturer at University of East Finland, interested in Mathematics Education)

Ismo Korkee, Ph. D., Juha Sillanpää, Lic. Phil.

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