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Hardcover ISBN:  9780821839638 
Product Code:  PSPUM/75 
List Price:  $139.00 
MAA Member Price:  $125.10 
AMS Member Price:  $111.20 
eBook ISBN:  9780821893814 
Product Code:  PSPUM/75.E 
List Price:  $135.00 
MAA Member Price:  $121.50 
AMS Member Price:  $108.00 
Hardcover ISBN:  9780821839638 
eBook ISBN:  9780821893814 
Product Code:  PSPUM/75.B 
List Price:  $274.00 $206.50 
MAA Member Price:  $246.60 $185.85 
AMS Member Price:  $219.20 $165.20 

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 ]