Preface
Published online by Cambridge University Press: 05 November 2011
Summary
Ever since Riemann's use of the theta transformation formula in one of his proofs of the functional equation for the zeta-function, number theorists have been fascinated by various interactions between the zeta function and automorphic forms. These experiences, however, have remained episodic like rare glimpses of crests, for most of them ensued from apparently spontaneous relations of the zeta-function with a variety of Eisenstein series. Nevertheless such glimpses are highly suggestive of a grand view over and far beyond the Eisenstein ridge, and bring forth the notion of a kamuy-mintar where the entire collection of automorphic forms contribute to the formation of the zeta-function.
My aim in the present monograph is to try to substantiate this belief by demonstrating that the zeta-function has indeed a structure tightly supported by all automorphic forms. The story begins with an unabridged treatment of the spectral resolution of the non-Euclidean Laplacian, and continues to a theory of trace formulas. The fundamental means thus readied are subsequently mustered up for the quest to find an explicit formula for the fourth power moment of the zeta-values. Then the zeta function emerges as a magnificent peak embracing infinitely many gems called automorphic L-functions representing the spectrum.
My best thanks are due to my friends A. Ivić and M. Jutila for their unfailing encouragement, and to D. Tranah, P. Jackson, and all of the personnel of the Cambridge University Press engaged in this project for sharing their professional vigor.
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- Publisher: Cambridge University PressPrint publication year: 1997