Bounded Gaps Between Primes

Searching for small gaps between consecutive primes is one way to approach the twin primes conjecture, one of the most celebrated unsolved problems in number theory. This book documents the remarkable developments of recent decades, whereby an upper bound on the known gap length between infinite numbers of consecutive primes has been reduced to a tractable finite size. The text is both introductory and complete: the detailed way in which results are proved is fully set out and plenty of background material is included. The reader journeys from selected historical theorems to the latest best result, exploring the contributions of a vast array of mathematicians, including Bombieri, Goldston, Motohashi, Pintz, Yildirim, Zhang, Maynard, Tao and Polymath8. The book is supported by a linked and freely-available package of computer programs. The material is suitable for graduate students and of interest to any mathematician curious about recent breakthroughs in the field.

• Lays out a path from the twentieth century to the best results of the twenty-first century, allowing readers to dip in and out
• Gives introductory and supporting background proofs and results including Bessel functions, compact operators, complex analysis and Weil's theorem for curves
• Includes software-engineered and documented computer programs so readers can attempt to improve the best results, for example by changing the code