Multiple Dirichlet Series, Automorphic Forms, and Analytic Number Theory

by Friedberg, Solomon; Bump, Daniel; Goldfeld, Dorian; Hoffstein, Jeffrey

• ISBN13: 9780821839638
• ISBN10: 0821839632
• Format: Hardcover
• Copyright: 2007-01-01
• Publisher: Amer Mathematical Society

Summary

Multiple Dirichlet series are Dirichlet series in several complex variables. A multiple Dirichlet series is said to be perfect if it satisfies a finite group of functional equations and has meromorphic continuation everywhere. The earliest examples came from Mellin transforms of metaplectic Eisenstein series and have been intensively studied over the last twenty years. More recently, many other examples have been discovered and it appears that all the classical theorems on moments of $L$-functions as well as the conjectures (such as those predicted by random matrix theory) can now be obtained via the theory of multiple Dirichlet series. Furthermore, new results, not obtainable by other methods, are just coming to light. This volume offers an account of some of the major research to date and the opportunities for the future. It includes an exposition of the main results in the theory of multiple Dirichlet series, and papers on moments of zeta- and $L$-functions, on new examples of multiple Dirichlet

Table of Contents

Multiple Dirichlet series and their applications: Multiple Dirichlet series and automorphic forms

Applications of multiple Dirichlet series in mean values of $L$-functions

Second moments of quadratic Hecke L-series and multiple Dirichlet series I

Weyl group multiple Dirichlet series I

Residues of Weyl group multiple Dirichlet series associated to $\widetilde{\mathrm{GL}}_{n+1}$

Multiple Hurwitz zeta functions

Multiple zeta values over global function fields

Generalised Selberg zeta functions and a conjectural Lefschetz formula

Automorphic forms and analytic number theory: Rankin-Cohen brackets on higher order modular forms

Eulerian integrals for $GL_n$

Is the Hlawka zeta function a respectable object?

On sums of integrals of powers of the zeta-function in short intervals

Uniform bounds for Rankin-Selberg $L$-functions

Mean values of zeta-functions via representation theory

On the pair correlation of the Eigenvalues of the hyperbolic Laplacian for PSL$(2,\mathbb{Z})\backslash H$ II

Lower bounds for moments of $L$-functions: Symplectic and orthogonal examples

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