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The Mahler measure is a fascinating notion and an exciting topic in contemporary mathematics, interconnecting with subjects as diverse as number theory, analysis, arithmetic geometry, special functions and random walks. This friendly and concise introduction to the Mahler measure is a valuable resource for both graduate courses and self-study. It provides the reader with the necessary background material, before presenting the recent achievements and the remaining challenges in the field. The first part introduces the univariate Mahler measure and addresses Lehmer's question, and then discusses techniques of reducing multivariate measures to hypergeometric functions. The second part touches on the novelties of the subject, especially the relation with elliptic curves, modular forms and special values of L-functions. Finally, the Appendix presents the modern definition of motivic cohomology and regulator maps, as well as Deligne-Beilinson cohomology. The text includes many exercises to test comprehension and challenge readers of all abilities.

Polynomials --- Measurement.

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Number Theory and Related Fields collects contributions based on the proceedings of the “International Number Theory Conference in Memory of Alf van der Poorten,” hosted by CARMA and held March 12–16, 2012, at the University of Newcastle, Australia. The purpose of the conference was to commemorate the research and influence of Alf van der Poorten in number theory and in general mathematics and presented an exciting venue for promoting number-theoretic research and graduate study in Australia. Comprehensive accounts of recent achievements in theoretical and computational number theory and its applications to cryptography and theoretical computer science were also paramount to the conference. The volume begins with a detailed academic appreciation of van der Poorten’s life and work, and includes research articles written by some of the most distinguished mathematicians in the field of number theory. Contributions also include related topics that focus on the various research interests of van der Poorten, such as continued fractions and elliptic curves. Researchers in number theory and its applications will find this Proceedings of great interest.

Number theory --- Mathematics --- Physical Sciences & Mathematics --- Algebra --- Number theory. --- Algebra. --- Number study --- Numbers, Theory of --- Mathematics. --- Number Theory. --- Math --- Science --- Mathematical analysis --- Van Der Poorten, A. J. --- Poorten, A. J. Van Der --- Der Poorten, A. J. van --- Van der Poorten, Alf --- Van der Poorten, Alfred J.

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Despite their classical nature, continued fractions are a neverending research area, with a body of results accessible enough to suit a wide audience, from researchers to students and even amateur enthusiasts. Neverending Fractions brings these results together, offering fresh perspectives on a mature subject. Beginning with a standard introduction to continued fractions, the book covers a diverse range of topics, from elementary and metric properties, to quadratic irrationals, to more exotic topics such as folded continued fractions and Somos sequences. Along the way, the authors reveal some amazing applications of the theory to seemingly unrelated problems in number theory. Previously scattered throughout the literature, these applications are brought together in this volume for the first time. A wide variety of exercises guide readers through the material, which will be especially helpful to readers using the book for self-study, and the authors also provide many pointers to the literature.

Continued fractions. --- Processes, Infinite. --- Fractions.

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We will describe a method for proving that a given real number is irrational. It amounts to constructing explicit rational approximants to the real number which are "better than possible" should the real number be rational. The rational approximants are obtained by evaluating a Hermite-Padé rational approximant to explicit functions at a convenient (integer) point. This constructive proof can also be used to prove linear independence over Q of some real numbers and in fact was first used by Hermite to prove the transcendence of e. The method is illustrated by various examples involving the zeta-function.