This volume marks the 20th anniversary of the New York Number Theory Seminar (NYNTS). Beginning in 1982, the NYNTS has tried to present a broad spectrum of research in number theory and related fields of mathematics, from physics to geometry to combinatorics and computer science. The list of seminar speakers includes not only Fields Medallists and other established researchers, but also many other younger and less well known mathematicians whose theorems are significant and whose work may become the next big thing in number theory.

* The spanning number and the independence number of a subset of an abelian group * A formula related to the Frobenius problem in two dimensions * One bit world * Use of Padé approximation in spline construction * Interactions between number theory and operator algebras in the study of Riemann zeta function (d'apres Bost-Connes and Connes) * A hyperelliptic curve with real multiplication of degree two * Humbert's conic model and the Kummer surface * Arithmeticity and theta correspondence of an orthogonal group * Morphis heights and periodic points * The elementary proof of the prime number theorem: An historical perspective * Additive bases representations and the Erdös-Túran conjecture * The boundary structure of the sumset in Z^2 * On NTU's in function fields * Continued fractions and quadratic irrationals * The inverse problem for representation functions of additive bases * On the ubiquity of Sidon sets