Region of Convergence of Dirichlet Series with Complex Exponents

In several recent papers G. L. Lunc (7, 8, 9, 10, 11) has examined the serieswhere {mn} is a sequence of positive integers or zeros, and μn > 0, μn ↑ ∞. The discussion has been aimed towards extending results which are well-known for Dirichlet series to series (1) which Lunc has very reasonably called a Taylor–Dirichlet series.

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HOMOTOPY ANALYSIS METHOD TO SOLVE BOUSSINESQ EQUATIONS

JOURNAL OF ADVANCES IN PHYSICS ◽

10.24297/jap.v10i3.1322 ◽

2015 ◽

Vol 10 (3) ◽

pp. 2825-2833

Author(s):

Achala Nargund ◽

R Madhusudhan ◽

S B Sathyanarayana

Keyword(s):

Homotopy Analysis Method ◽

Degrees Of Freedom ◽

Initial Approximation ◽

Boussinesq Equations ◽

Convergent Series ◽

Boussinesq System ◽

Analysis Method ◽

Analytical Technique ◽

Homotopy Analysis ◽

Region Of Convergence

In this paper, Homotopy analysis method is applied to the nonlinear coupleddifferential equations of classical Boussinesq system. We have applied Homotopy analysisÂ method (HAM) for the application problems in [1, 2, 3, 4]. We have also plotted Domb-SykesÂ plot for the region of convergence. We have applied Pade for the HAM series to identify theÂ singularity and reflect it in the graph. The HAM is a analytical technique which is used toÂ solve non-linear problems to generate a convergent series. HAM gives complete freedom toÂ choose the initial approximation of the solution, it is the auxiliary parameter h which gives us aÂ convenient way to guarantee the convergence of homotopy series solution. It seems that moreartificial degrees of freedom implies larger possibility to gain better approximations by HAM.

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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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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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