Research Directions in Number Theory

Number theory is one of the oldest and most appealing areas of mathematics. Computation has always played a role in number theory, a role which has increased dramatically in the last 20 or 30 years, both because of the advent of modern computers, and because of the discovery of surprising and powerful algorithms.

The subject of the book is Diophantine approximation and Nevanlinna theory. Not only does the text provide new results and directions, it also challenges open problems and collects latest research activities on these subjects made by the authors over the past eight years. Some of the significant findings are the proof of the Green-Griffiths conjecture by using meromorphic connections and Jacobian sections, and a generalized abc-conjecture.

This book is a survey of the most important directions of research in transcendental number theory. The central topics in this theory include proofs of irrationality and transcendence of various numbers, especially those that arise as the values of special functions. Questions of this sort go back to ancient times. An example is the old problem of squaring the circle, which Lindemann showed to be impossible in 1882, when he proved that $Öpi$ is a transcendental number. Euler's conjecture that the logarithm of an algebraic number to an algebraic base is transcendental was included in Hilbert's famous list of open problems; this conjecture was proved by Gel'fond and Schneider in 1934.