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Book DetailsProceedings of Symposia in Pure MathematicsVolume: 75; 2006; 303 ppMSC: Primary 11; Secondary 22;
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 series, and on developments in the allied fields of automorphic forms and analytic number theory.
ReadershipGraduate students and research mathematicians interested in number theory, especially automorphic forms and connections to analytic number theory.

Table of Contents

Multiple Dirichlet series and their applications

Gautam Chinta, Solomon Friedberg and Jeffrey Hoffstein  Multiple Dirichlet series and automorphic forms [ MR 2279929 ]

Qiao Zhang  Applications of multiple Dirichlet series in mean values of $L$functions [ MR 2279930 ]

Adrian Diaconu and Dorian Goldfeld  Second moments of quadratic Hecke $L$series and multiple Dirichlet series I [ MR 2279931 ]

Benjamin Brubaker, Daniel Bump, Gautam Chinta, Solomon Friedberg and Jeffrey Hoffstein  Weyl group multiple Dirichlet series I [ MR 2279932 ]

Benjamin Brubaker and Daniel Bump  Residues of Weyl group multiple Dirichlet series associated to $\widetilde {\mathrm {GL}}_{n+1}$ [ MR 2279933 ]

M. Ram Murty and Kaneenika Sinha  Multiple Hurwitz zeta functions [ MR 2279934 ]

Riad Masri  Multiple zeta values over global function fields [ MR 2279935 ]

Anton Deitmar  Generalised Selberg zeta functions and a conjectural Lefschetz formula [ MR 2279936 ]

Automorphic forms and analytic number theory

Y. Choie and N. Diamantis  RankinCohen brackets on higher order modular forms [ MR 2279937 ]

David Ginzburg  Eulerian integrals for $\mathrm {GL}_n$ [ MR 2279938 ]

M. N. Huxley  Is the Hlawka zeta function a respectable object? [ MR 2279939 ]

Aleksandar Ivić  On sums of integrals of powers of the zetafunction in short intervals [ MR 2279940 ]

Matti Jutila and Yoichi Motohashi  Uniform bounds for RankinSelberg $L$functions [ MR 2279941 ]

Yoichi Motohashi  Mean values of zetafunctions via representation theory [ MR 2279942 ]

C. J. Mozzochi  On the pair correlation of the eigenvalues of the hyperbolic Laplacian for $\mathrm {PSL}(2, \mathbb {Z})\backslash H$. II [ MR 2279943 ]

Z. Rudnick and K. Soundararajan  Lower bounds for moments of $L$functions: symplectic and orthogonal examples [ MR 2279944 ]


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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 series, and on developments in the allied fields of automorphic forms and analytic number theory.
Graduate students and research mathematicians interested in number theory, especially automorphic forms and connections to analytic number theory.

Multiple Dirichlet series and their applications

Gautam Chinta, Solomon Friedberg and Jeffrey Hoffstein  Multiple Dirichlet series and automorphic forms [ MR 2279929 ]

Qiao Zhang  Applications of multiple Dirichlet series in mean values of $L$functions [ MR 2279930 ]

Adrian Diaconu and Dorian Goldfeld  Second moments of quadratic Hecke $L$series and multiple Dirichlet series I [ MR 2279931 ]

Benjamin Brubaker, Daniel Bump, Gautam Chinta, Solomon Friedberg and Jeffrey Hoffstein  Weyl group multiple Dirichlet series I [ MR 2279932 ]

Benjamin Brubaker and Daniel Bump  Residues of Weyl group multiple Dirichlet series associated to $\widetilde {\mathrm {GL}}_{n+1}$ [ MR 2279933 ]

M. Ram Murty and Kaneenika Sinha  Multiple Hurwitz zeta functions [ MR 2279934 ]

Riad Masri  Multiple zeta values over global function fields [ MR 2279935 ]

Anton Deitmar  Generalised Selberg zeta functions and a conjectural Lefschetz formula [ MR 2279936 ]

Automorphic forms and analytic number theory

Y. Choie and N. Diamantis  RankinCohen brackets on higher order modular forms [ MR 2279937 ]

David Ginzburg  Eulerian integrals for $\mathrm {GL}_n$ [ MR 2279938 ]

M. N. Huxley  Is the Hlawka zeta function a respectable object? [ MR 2279939 ]

Aleksandar Ivić  On sums of integrals of powers of the zetafunction in short intervals [ MR 2279940 ]

Matti Jutila and Yoichi Motohashi  Uniform bounds for RankinSelberg $L$functions [ MR 2279941 ]

Yoichi Motohashi  Mean values of zetafunctions via representation theory [ MR 2279942 ]

C. J. Mozzochi  On the pair correlation of the eigenvalues of the hyperbolic Laplacian for $\mathrm {PSL}(2, \mathbb {Z})\backslash H$. II [ MR 2279943 ]

Z. Rudnick and K. Soundararajan  Lower bounds for moments of $L$functions: symplectic and orthogonal examples [ MR 2279944 ]