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The Riemann hypothesis (RH) may be the most important outstanding problem in mathematics. This third volume on equivalents to RH comprehensively presents recent results of Nicolas, Rogers¿Tao¿Dobner, Polymath15, and Matiyasevich. Particularly interesting are derivations which show, assuming all zeros on the critical line are simple, that RH is decidable. Also included are classical PólyäJensen equivalence and related developments of Ono et al. Extensive appendices highlight key background results, most of which are proved. The book is highly accessible, with definitions repeated, proofs split logically, and graphical visuals. It is ideal for mathematicians wishing to update their knowledge, logicians, and graduate students seeking accessible number theory research problems. The three volumes can be read mostly independently. Volume 1 presents classical and modern arithmetic RH equivalents. Volume 2 covers equivalences with a strong analytic orientation. Volume 3 includes further arithmetic and analytic equivalents plus new material on RH decidability.

Constructive mathematics ¿ mathematics in which 'there exists' always means 'we can construct' ¿ is enjoying a renaissance. fifty years on from Bishop's groundbreaking account of constructive analysis, constructive mathematics has spread out to touch almost all areas of mathematics and to have profound influence in theoretical computer science. This handbook gives the most complete overview of modern constructive mathematics, with contributions from leading specialists surveying the subject's myriad aspects. Major themes include: constructive algebra and geometry, constructive analysis, constructive topology, constructive logic and foundations of mathematics, and computational aspects of constructive mathematics. A series of introductory chapters provides graduate students and other newcomers to the subject with foundations for the surveys that follow. Edited by four of the most eminent experts in the field, this is an indispensable reference for constructive mathematicians and a fascinating vista of modern constructivism for the increasing number of researchers interested in constructive approaches.

This book, targeted at graduate students and researchers interested in functional analysis, gives a comprehensive coverage of classic and recent works on the numerical range theory. With detailed references to the literature and numerous exercises, it serves as an accessible entry point into this lively and exciting research area.