Laurent expansion of Dirichlet series-II

U. Balakrishnan

doi:10.1007/bf01302705

Stieltjes constants of L-functions in the extended Selberg class

The Ramanujan Journal ◽

10.1007/s11139-021-00391-1 ◽

2021 ◽

Author(s):

Shōta Inoue ◽

Sumaia Saad Eddin ◽

Ade Irma Suriajaya

Keyword(s):

Dirichlet Series ◽

Upper Bound ◽

Arithmetic Function ◽

Laurent Expansion ◽

Selberg Class ◽

Stieltjes Constants ◽

Extended Selberg Class

AbstractLet f be an arithmetic function and let $${\mathcal {S}}^\#$$ S # denote the extended Selberg class. We denote by $${\mathcal {L}}(s) = \sum _{n = 1}^{\infty }\frac{f(n)}{n^s}$$ L ( s ) = ∑ n = 1 ∞ f ( n ) n s the Dirichlet series attached to f. The Laurent–Stieltjes constants of $${\mathcal {L}}(s)$$ L ( s ) , which belongs to $${\mathcal {S}}^\#$$ S # , are the coefficients of the Laurent expansion of $${\mathcal {L}}$$ L at its pole $$s=1$$ s = 1 . In this paper, we give an upper bound of these constants, which is a generalization of many known results.

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Laurent expansion of Dirichlet series

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700003919 ◽

1986 ◽

Vol 33 (3) ◽

pp. 351-357 ◽

Cited By ~ 3

Author(s):

U. Balakrishnan

Keyword(s):

Dirichlet Series ◽

Laurent Expansion ◽

Real Numbers ◽

Positive Real ◽

Abscissa Of Convergence ◽

Image Position

Let 〈an〉 be an increasing sequence of real numbers and 〈bn a sequence of positive real numbers. We deal here with the Dirichlet series and its Laurent expansion at the abscissa of convergence, λ, say. When an and bn behave likeas N → ∞, where P2(x) is a certain polynomial, we obtain the Laurent expansion of f (s) at s = λ, namelywhere P1(x) is a polynomial connected with P2(x) above. Also, the connection between P1 and P2 is made intuitively transparent in the proof.

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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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