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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.
This volume presents an authoritative, up-to-date review of
analytic number theory. It contains outstanding contributions from
leading international figures in this field. Core topics discussed
include the theory of zeta functions, spectral theory of
automorphic forms, classical problems in additive number theory
such as the Goldbach conjecture, and diophantine approximations and
equations. This will be a valuable book for graduates and
researchers working in number theory.
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.
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