Riemann’s zeta function and beyond

DOI10.1090/S0273-0979-03-00995-9zbMath1046.11001arXivmath/0309478OpenAlexW1969448553MaRDI QIDQ4452146

Stephen D. Miller, Stephen Gelbart

Publication date: 12 February 2004

Published in: Bulletin of the American Mathematical Society (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/math/0309478

zbMATH Keywords

\(L\)-functionsautomorphic formsthe Langlands Program

Mathematics Subject Classification ID

(zeta (s)) and (L(s, chi)) (11M06) Research exposition (monographs, survey articles) pertaining to number theory (11-02) Other Dirichlet series and zeta functions (11M41) Langlands (L)-functions; one variable Dirichlet series and functional equations (11F66) Special classes of entire functions of one complex variable and growth estimates (30D15)

Related Items

Aspects of Zeta-Function Theory in the Mathematical Works of Adolf Hurwitz ⋮ Some analytic continuations of the Barnes zeta function in two and higher dimensions ⋮ Periods of automorphic forms, poles of \(L\)-functions and functorial lifting ⋮ Twisting moduli for \(\mathrm{GL}(2)\) ⋮ Li coefficients for automorphic \(L\)-functions ⋮ Eisenstein series on affine Kac-Moody groups over function fields ⋮ Finite and infinite order differential relations for theta functions ⋮ Zeta functions on tori using contour integration ⋮ On a relation between modular functions and Dirichlet series: found in the estate of Adolf Hurwitz ⋮ Converse theorems for automorphic distributions and Maass forms of level \(N\) ⋮ Central values of additive twists of cuspidal \(L\)-functions ⋮ ANALYTIC CONTINUATION OF NONANALYTIC VECTOR-VALUED EISENSTEIN SERIES ⋮ Weil’s converse theorem for Maass forms and cancellation of zeros

Cites Work

• Unnamed Item
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• Converse theorems for GLn, II
• Langlands-Shahidi Method and Poles of Automorphic L-Functions: Application to Exterior Square L-Functions
• Une caractérisation de la correspondance de Langlands locale pour ${\rm GL}(n)$
• Spectra of hyperbolic surfaces
• Recent advances in the Langlands Program
• Boundedness of automorphic 𝐿–functions in vertical strips
• Functorial products for GL2×GL3 and functorial symmetric cube for GL2
• The weyl theorem and the deformation of discrete groups
• The principle of functoriality
• Relative modular symbols and Rankin-Selberg convolutions
• Rankin-Selberg Integrals for the Group G 2
• Notes on elliptic curves. II.
• Decomposition of Eisenstein series: Rankin triple products
• On cusp forms for co-finite subgroups of PSL(2,\({\mathbb{R}})\)
• Automorphic forms on GL(3,\({\mathbb{R}})\)
• The central critical value of a triple product \(L\)-function
• The trace class conjecture in the theory of automorphic forms
• Fourier coefficients of modular forms of half-integral weight
• Heegner points and derivatives of \(L\)-series
• Hyperbolic distribution problems and half-integral weight Maass forms
• The symmetric-square L-function attached to a cuspidal automorphic representation of \(GL_ 3\)
• On the Ramanujan conjecture and finiteness of poles for certain \(L\)-functions
• Ramanujan graphs
• Local coefficients as Artin factors for real groups
• Symmetric square \(L\)-functions on \(GL(r)\)
• Spectral limits for hyperbolic surfaces. I
• Bounds for automorphic \(L\)-functions
• The multiplicity one theorem for \(\mathrm{GL}_n\)
• On the functional equations satisfied by Eisenstein series
• A non-vanishing theorem for zeta functions of \(\mathrm{GL}_n\)
• Automorphic forms on \(GL(3)\). I, II
• Disappearance of cusp forms in special families
• Discrete groups, expanding graphs and invariant measures. Appendix by Jonathan D. Rogawski
• Converse theorems for \(\text{GL}_n\)
• On the coefficients of cusp forms
• Rational points on the sphere
• Poles of Artin \(L\)-functions and the strong Artin conjecture
• Modularity of the Rankin-Selberg \(L\)-series, and multiplicity one for \(\mathrm{SL}(2)\)
• Drinfeld shtukas and Langlands correspondence.
• Spectrum and homology of congruence complex hyperbolic manifolds
• A proof of Langlands' conjecture on Plancherel measures; complementary series for \(p\)-adic groups
• On the local Langlands conjecture for \(\text{GL}_n\)
• Sums of Kloosterman sums
• La conjecture de Weil. I
• Cuspidality of symmetric powers with applications.
• Fourier coefficients of modular forms on \(G_2\).
• On lifting from classical groups to \(\text{GL}_N\)
• Functoriality for the classical groups
• Distributions and analytic continuation of Dirichlet series
• Modular elliptic curves and Fermat's Last Theorem
• Ring-theoretic properties of certain Hecke algebras
• On Selberg's eigenvalue conjecture
• A converse theorem for \(GL_ 4\)
• Algebraic modular forms
• Symmetric cube \(L\)-functions for \(\text{GL}_2\) are entire
• On the gamma factor of the triple \(L\)-function. I
• A simple proof of Langlands conjectures for \(\text{GL}_n\) on a \(p\)-adic field
• Über die Bestimmung Dirichletscher Reihen durch Funktionalgleichungen
• Automorphic forms on semisimple Lie groups. Notes by J. G. M. Mars.
• A remark on the Sato-Tate conjecture
• Automorphic forms on GL (2)
• Zeta functions of simple algebras
• Über eine neue Art von nichtanalytischen automorphen Funktionen und die Bestimmung Dirichletscher Reihen durch Funktionalgleichungen. (On a new type of nonanalytic automorphic functions and the determination of Dirichlet series by functional equations)
• On the modularity of elliptic curves over 𝐐: Wild 3-adic exercises
• The Trace Formula and Its Applications: An Introduction to the Work of James Arthur
• A Proof of Casselman-Shahidi’s Conjecture for Quasi-split Classical Groups
• The Geometry and Cohomology of Some Simple Shimura Varieties. (AM-151)
• Applications of Langlands’ functorial lift of odd orthogonal groups
• Rankin-Selberg Convolutions
• An elementary introduction to the Langlands program
• 𝐿 functions for the group 𝐺₂
• Analytic Proof of the Strong Multiplicity One Theorem
• A new way to get Euler products.
• On Certain L-Functions
• Artin’s conjecture for representations of octahedral type
• On Euler Products and the Classification of Automorphic Forms II
• Perturbation Theory for the Laplacian on Automorphic Functions
• Automorphic Representation of Unitary Groups in Three Variables. (AM-123)
• On the Holomorphy of Certain Dirichlet Series
• Automorphic Forms on Adele Groups. (AM-83)
• Modular Forms for Γ 0 (N) and Dirichlet Series
• On an explicit formula for class-$1$ “Whittaker functions” on $GL_n $ over $\beta $-adic fields
• A relation between automorphic representations of ${\rm GL}(2)$ and ${\rm GL}(3)$
• A converse theorem for Dirichlet series with poles