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.
We prove the Kloosterman-Spectral sum formula for PSL 2 (Z[i])\PSL 2 (C), and apply it to derive an explicit spectral expansion for the fourth power moment of the Dedekind zeta function of the Gaussian number field. Our sum formula, Theorem 13.1, allows the extension of the spectral theory of Kloosterman sums to all algebraic number fields. 2 −δ+it g(t) Γ(δ − it) ∞ 0 r 2(s−γ)−δ−it−1 (2.16) × ∞ 0 y δ−it−1 e − 1 2 y(r+1/r) I |q| (y)dy dr dt = (−1) q ∞ −∞ 2 1−δ+it g(t) Γ(δ − it) × ∞ 0 y δ−it−1 K 2(s−γ)−δ−it (y)I |q| (y)dy dt,
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.