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Four Faces of Number Theory
EMS Series of Lectures in Mathematics

Kathrin Bringmann (Universität zu Köln, Germany)
Yann Bugeaud (IRMA Strasbourg, France)
Titus Hilberdink (University of Reading, UK)
Jürgen W. Sander (Universität Hildesheim, Germany)

Four Faces of Number Theory

ISBN print 978-3-03719-142-2, ISBN online 978-3-03719-642-7
DOI 10.4171/142
November 2015, 198 pages, softcover, 17 x 24 cm.
32.00 Euro

This book arose from courses given at the International Summer School organized in August 2012 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 (under)graduate 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.

Keywords: Transcendence, algebraic number, Schmidt Subspace Theorem, continued fraction, digital expansion, Diophantine approximation