# Four Faces of Number Theory

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*Kathrin Bringmann; Yann Bugeaud; Titus Hilberdink; Jürgen Sander*

A publication of the European Mathematical Society

This book arose from courses given at an International Summer School
organized by the number theory group of the Department of Mathematics at
the University of Würzburg. It consists of four essentially
self-contained chapters and presents recent research results
highlighting the strong interplay between number theory and other fields
of mathematics, such as combinatorics, functional analysis and graph
theory. The book is addressed to undergraduate students who wish to
discover various aspects of number theory. Remarkably, it demonstrates
how easily one can approach frontiers of current research in number
theory by elementary and basic analytic methods.

Kathrin Bringmann gives an introduction to the theory of modular forms
and, in particular, so-called Mock theta-functions, a topic which had
been untouched for decades but has obtained much attention in the last
years. Yann Bugeaud is concerned with expansions of algebraic numbers.
Here combinatorics on words and transcendence theory are combined to
derive new information on the sequence of decimals of algebraic numbers
and on their continued fraction expansions. Titus Hilberdink reports on
a recent and rather unexpected approach to extreme values of the Riemann
zeta-function by use of (multiplicative) Toeplitz matrices and
functional analysis. Finally, Jürgen Sander gives an introduction
to algebraic graph theory and the impact of number theoretical methods
on fundamental questions about the spectra of graphs and the analogue of
the Riemann hypothesis.

A publication of the European Mathematical Society (EMS). Distributed within the Americas by the American Mathematical Society.

#### Readership

Undergraduate and graduate students interested in number theory.