Kirjahaku

The Riemann zeta function is one of the most studied objects in mathematics, and is of fundamental importance. In this book, based on his own research, Professor Motohashi shows that the function is closely bound with automorphic forms and that many results from there can be woven with techniques and ideas from analytic number theory to yield new insights into, and views of, the zeta function itself. The story starts with an elementary but unabridged treatment of the spectral resolution of the non-Euclidean Laplacian and the trace formulas. This is achieved by the use of standard tools from analysis rather than any heavy machinery, forging a substantial aid for beginners in spectral theory as well. These ideas are then utilized to unveil an image of the zeta-function, first perceived by the author, revealing it to be the main gem of a necklace composed of all automorphic L-functions. In this book, readers will find a detailed account of one of the most fascinating stories in the development of number theory, namely the fusion of two main fields in mathematics that were previously studied separately.

The Riemann zeta function is one of the most studied objects in mathematics, and is of fundamental importance. In this book, based on his own recent research, Professor Motohashi shows that the function is closely bound with automorphic forms and that many results from there can be woven with techniques and ideas from analytic number theory to yield new insights into, and views of, the zeta function itself. The story starts with an elementary but unabridged treatment of the spectral resolution of the non-Euclidean Laplacian and the trace formulas. This is achieved by the use of standard tools from analysis rather than any heavy machinery, forging a substantial aid for beginners in spectral theory as well. These ideas are then utilised to unveil a new image of the zeta-function, first perceived by the author, revealing it to be the main gem of a necklace composed of all automorphic L-functions. In this book, readers will find a detailed account of one of the most fascinating stories in the recent development of number theory, namely the fusion of two main fields in mathematics that were previously studied separately.

Offering a comprehensive introduction to number theory, this is the ideal book both for those who want to learn the subject seriously and independently, or for those already working in number theory who want to deepen their expertise. Readers will be treated to a rich experience, developing the key theoretical ideas while explicitly solving arithmetic problems, with the historical background of analytic and algebraic number theory woven throughout. Topics include methods of solving binomial congruences, a clear account of the quantum factorization of integers, and methods of explicitly representing integers by quadratic forms over integers. In the later parts of the book, the author provides a thorough approach towards composition and genera of quadratic forms, as well as the essentials for detecting bounded gaps between prime numbers that occur infinitely often.