Summation formulas, from Poisson and Voronoi to the present

Miller, Stephen D.; Schmid, Wilfried

doi:10.1007/978-0-8176-8204-0_15

Cited by 37 publications

(34 citation statements)

References 17 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Similar formulas have found many applications in L-functions theory, most notably in the proofs of subconvexity results. They are all called formulas of Voronoi type, and we refer to [8] for a general survey of recent developments. The summation formula for τ k is obtained by Ivić in [3], and we will quote here some of his results that we will need.…”

Section: A Summation Formula For the Twisted Divisor Functionmentioning

confidence: 99%

The sixth moment of the family of Γ1(q)-automorphic L-functions

Djanković

2011

Arch. Math.

View full text Add to dashboard Cite

show abstract

Section: A Summation Formula For the Twisted Divisor Functionmentioning

confidence: 99%

The sixth moment of the family of Γ1(q)-automorphic L-functions

Djanković

2011

Arch. Math.

View full text Add to dashboard Cite

show abstract

“…The comment following (6.55) applies triply in the current setting. The introduction to [7] sketches a proof the Voronoi summation formula for SL (2), which has a long history [6]. Our formulation involves the SL(2) analogue of the integral transform (6.51),…”

Section: (S) Has Locally Uniform Polynomial Growth On Vertical Linesmentioning

confidence: 99%

Distributions and analytic continuation of Dirichlet series

Miller

Schmid

2004

Journal of Functional Analysis

Self Cite

View full text Add to dashboard Cite

This is the second in a series of three papers; the other two are "Summation Formulas, from Poisson and Voronoi to the Present" [Progr. Math. 220 (2004) and "Automorphic Distributions, L-functions, and Voronoi Summation for GL(3)" (preprint). The first paper is primarily expository, while the third proves a Voronoi-style summation formula for the coefficients of a cusp form on GL(3, Z)\GL(3, R). The present paper contains the distributional machinery used in the third paper for rigorously deriving the summation formula, and also for the proof of the GL(3) × GL(1) converse theorem given in the third paper. The primary concept studied is a notion of the order of vanishing of a distribution along a closed submanifold. Applications are given to the analytic continuation of Riemann's zeta function, degree 1 and degree 2 L-functions, the converse theorem for GL(2), and a characterization of the classical Mellin transform/inversion relations on functions with specified singularities.

show abstract

“…This essential insight explains exactly the reason for the appearance of (6). Another representation-theoretic approach, including a Voronoı¨summation formula for GLð3Þ is taken by Miller and Schmid [25,26].…”

Section: Introductionmentioning

confidence: 99%

“…These coefficients are, like tðnÞ; the Fourier coefficients of a modular form. See Berndt [4], Miller and Schmid [26], and Wilton [33], for references to the literature of this problem.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Moments of the Riemann zeta function and Eisenstein series—I

Beineke¹,

Bump²

2004

Journal of Number Theory

View full text Add to dashboard Cite

It is shown that if the parameters of an Eisenstein series on GLð2kÞ are chosen so that its (integrated) L-function is the 2kth moment of the Riemann zeta function, then the 2k k À Á terms in its constant term agree with 2k k À Á factors appearing in a conjectural formula for the 2kth moment of zeta by Conrey, Farmer, Keating, Rubinstein and Snaith. When k ¼ 1; an explanation for this phenomenon is found by deducing Oppenheim's generalization of the Voronoı¨summation formula from the Eisenstein series and representation theoretic considerations. The possibility of eliminating the problematical ''arithmetic factor'' is discussed. r

show abstract

Summation formulas, from Poisson and Voronoi to the present

Cited by 37 publications

References 17 publications

The sixth moment of the family of Γ1(q)-automorphic L-functions

The sixth moment of the family of Γ1(q)-automorphic L-functions

Distributions and analytic continuation of Dirichlet series

Moments of the Riemann zeta function and Eisenstein series—I

Contact Info

Product

Resources

About