A Dirichlet Series of Eigenvalue Type with Applications to Asymptotic Estimates

We establish asymptotic estimates of Mathieu-type series defined by sequences with power-logarithmic or factorial behavior. By taking the Mellin transform, the problem is mapped to the singular behavior of certain Dirichlet series, which is then translated into asymptotics for the original series. In the case of power-logarithmic sequences, we obtain precise first order asymptotics. For factorial sequences, a natural boundary of the Mellin transform makes the problem more challenging, but a direct elementary estimate gives reasonably precise asymptotics. As a byproduct, we prove an expansion of the functional inverse of the gamma function at infinity.

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On Dirichlet Series Attached to Holomorphic Cusp Forms on $SO(2, q)$

Automorphic Forms and Number Theory ◽

10.2969/aspm/00710333 ◽

2018 ◽

Cited By ~ 3

Author(s):

Takashi Sugano

Keyword(s):

Dirichlet Series ◽

Cusp Forms ◽

Holomorphic Cusp Forms

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On a family of weighted convolution algebras

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/s0161171290000758 ◽

1990 ◽

Vol 13 (3) ◽

pp. 517-525 ◽

Cited By ~ 9

Author(s):

Hans G. Feichtinger ◽

A. Turan Gürkanli

Keyword(s):

Fourier Transforms ◽

Locally Compact Abelian Group ◽

Asymptotic Estimates ◽

Locally Compact ◽

Invariance Properties ◽

Beurling Algebra ◽

Convolution Algebras ◽

Approximate Identities ◽

Beurling Algebras ◽

Banach Ideals

Continuing a line of research initiated by Larsen, Liu and Wang [12], Martin and Yap [13], Gürkanli [15], and influenced by Reiter's presentation of Beurling and Segal algebras in Reiter [2,10] this paper presents the study of a family of Banach ideals of Beurling algebrasLw1(G),Ga locally compact Abelian group. These spaces are defined by weightedLp-conditions of their Fourier transforms. In the first section invariance properties and asymptotic estimates for the translation and modulation operators are given. Using these it is possible to characterize inclusions in section 3 and to show that two spaces of this type coincide if and only if their parameters are equal. In section 4 the existence of approximate identities in these algebras is established, from which, among other consequences, the bijection between the closed ideals of these algebras and those of the corresponding Beurling algebra is derived.

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The existence, uniqueness and asymptotic estimates of solutions for third-order full nonlinear singularly perturbed vector boundary value problems

Boundary Value Problems ◽

10.1186/s13661-020-01322-7 ◽

2020 ◽

Vol 2020 (1) ◽

Author(s):

Xiaojie Lin ◽

Jiang Liu ◽

Can Wang

Keyword(s):

Boundary Value Problems ◽

Boundary Value ◽

Singularly Perturbed ◽

Asymptotic Estimates ◽

Third Order ◽

Estimates Of Solutions

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Cancellation in algebraic twisted sums on GL_ m

Forum Mathematicum ◽

10.1515/forum-2020-0252 ◽

2021 ◽

Vol 33 (4) ◽

pp. 1061-1082

Author(s):

Yujiao Jiang ◽

Guangshi Lü

Keyword(s):

Dirichlet Series ◽

Central Character ◽

Multiplicative Functions ◽

Cuspidal Representation ◽

Series Coefficient

Abstract Let π be an automorphic irreducible cuspidal representation of GL m {\operatorname{GL}_{m}} over ℚ {\mathbb{Q}} with unitary central character, and let λ π ⁢ ( n ) {\lambda_{\pi}(n)} be its n-th Dirichlet series coefficient. We study short sums of isotypic trace functions associated to some sheaves modulo primes q of bounded conductor, twisted by multiplicative functions λ π ⁢ ( n ) {\lambda_{\pi}(n)} and μ ⁢ ( n ) ⁢ λ π ⁢ ( n ) {\mu(n)\lambda_{\pi}(n)} . We are able to establish non-trivial bounds for these algebraic twisted sums with intervals of length of at least q 1 / 2 + ε {q^{1/2+\varepsilon}} for an arbitrary fixed ε > 0 {\varepsilon>0} .

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A dominated convergence theorem for Eisenstein series

Annales mathématiques du Québec ◽

10.1007/s40316-021-00157-7 ◽

2021 ◽

Author(s):

Johann Franke

Keyword(s):

Fourier Series ◽

Dirichlet Series ◽

Eisenstein Series ◽

Modular Forms ◽

Convergence Theorem ◽

Rational Functions ◽

New Approach ◽

Series Representations ◽

Dominated Convergence ◽

Generalized Dirichlet

AbstractBased on the new approach to modular forms presented in [6] that uses rational functions, we prove a dominated convergence theorem for certain modular forms in the Eisenstein space. It states that certain rearrangements of the Fourier series will converge very fast near the cusp $$\tau = 0$$ τ = 0 . As an application, we consider L-functions associated to products of Eisenstein series and present natural generalized Dirichlet series representations that converge in an expanded half plane.

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