Estimates for Sums of Coefficients of Dirichlet Series with Functional Equation

2003 ◽  
pp. 223-233
Author(s):  
V. Kumar Murty
Keyword(s):  
Functional Equation ◽  
Dirichlet Series

scholarly journals A note on a functional equation

1973 ◽  
Vol 15 (4) ◽  
pp. 385-388
Author(s):  
Chung-Ming An
Keyword(s):  
Functional Equation ◽  
Dirichlet Series ◽  
Gamma Function ◽  
Generalized Gamma ◽  
Special Cases ◽  
Generalized Gamma Function

The object of this note is to give an aspect to the problem of the functional equation of the generalized gamma function and Dirichlet series which are defined in [1]. In general, we cannot answer the problem yet. But it is worthy to attack this problem for some special cases.

SOME NUMBER THEORETIC APPLICATIONS OF A GENERAL MODULAR RELATION

2006 ◽  
Vol 02 (04) ◽  
pp. 599-615 ◽  
Author(s):  
SHIGERU KANEMITSU ◽  
YOSHIO TANIGAWA ◽  
HARUO TSUKADA
Keyword(s):  
Functional Equation ◽  
Dirichlet Series ◽  
The Other ◽  
Gamma Functions ◽  
Modular Relation

We state a form of the modular relation in which the functional equation appears in the form of an expression of one Dirichlet series in terms of the other multiplied by the quotient of gamma functions and illustrate it by some concrete examples including the results of Koshlyakov, Berndt and Wigert and Bellman.

SOLVING LINEAR EQUATIONS IN L-FUNCTIONS

2014 ◽  
Vol 10 (03) ◽  
pp. 569-584
Author(s):  
J. KACZOROWSKI ◽  
A. PERELLI
Keyword(s):  
Functional Equation ◽  
Linear Equation ◽  
Dirichlet Series ◽  
Linear Equations

We describe the solutions of the linear equation aX + bY = cZ in the class of Dirichlet series with functional equation. Proofs are based on the properties of certain nonlinear twists of the L-functions.

A Hecke correspondence theorem for automorphic integrals with infinite log-polynomial sum period functions

2014 ◽  
Vol 10 (07) ◽  
pp. 1857-1879 ◽  
Author(s):  
Austin Daughton
Keyword(s):  
Functional Equation ◽  
Dirichlet Series ◽  
Vertical Strip ◽  
Hecke Groups ◽  
Hecke Correspondence ◽  
Correspondence Theorem ◽  
Automorphic Integrals

We generalize the correspondence between Dirichlet series with finitely many poles that satisfy a functional equation and automorphic integrals with log-polynomial sum period functions. In particular, we extend the correspondence to hold for Dirichlet series with finitely many essential singularities. We also study Dirichlet series with infinitely many poles in a vertical strip. For Hecke groups with λ ≥ 2 and some weights, we prove a similar correspondence for these Dirichlet series. For this case, we provide a way to estimate automorphic integrals with infinite log-polynomial periods by automorphic integrals with finite log-polynomial periods.

scholarly journals A generalization of Bochner's formula

2004 ◽  
Vol Volume 27 ◽  
Author(s):  
S Kanemitsu ◽  
Y Tanigawa ◽  
H Tsukada
Keyword(s):  
Functional Equation ◽  
Dirichlet Series ◽  
Hierarchy Theorems ◽  
International Audience ◽  
Natural Way

International audience In this note we expound our general hierarchy theorems by the example of a Ramified-Type Functional Equarion H, which gives all possbile forms, in terms of se-ries with H-function coefficients, of the functional equation of higher hierarchy arising from the original ramified one satisfied by the Dirichlet series. Then by sepcifying the parameters, we shall deduce a few concrete examples scattered in the literature in the most natural way.

scholarly journals Generalised Dirichlet series and Hecke's functional equation

1967 ◽  
Vol 15 (4) ◽  
pp. 309-313 ◽  
Author(s):  
Bruce C. Berndt
Keyword(s):  
Functional Equation ◽  
Zeta Function ◽  
Dirichlet Series ◽  
Riemann Zeta Function ◽  
General Class ◽  
The Riemann Zeta Function ◽  
Entire Complex ◽  
Riemann Zeta ◽  
Image Position

The generalised zeta-function ζ(s, α) is defined bywhere α>0 and Res>l. Clearly, ζ(s, 1)=, where ζ(s) denotes the Riemann zeta-function. In this paper we consider a general class of Dirichlet series satisfying a functional equation similar to that of ζ(s). If ø(s) is such a series, we analogously define ø(s, α). We shall derive a representation for ø(s, α) which will be valid in the entire complex s-plane. From this representation we determine some simple properties of ø(s, α).

Summation Formulae for Coefficients of L-functions

2005 ◽  
Vol 57 (3) ◽  
pp. 494-505 ◽  
Author(s):  
John B. Friedlander ◽  
Henryk Iwaniec
Keyword(s):  
Functional Equation ◽  
Dirichlet Series ◽  
Summation Formula ◽  
General Dirichlet Series ◽  
Summation Formulae

AbstractWith applications inmind we establish a summation formula for the coefficients of a general Dirichlet series satisfying a suitable functional equation. Among a number of consequences we derive a generalization of an elegant divisor sum bound due to F. V. Atkinson.

scholarly journals Subconvexity for a double Dirichlet series and non-vanishing of L-functions

2018 ◽  
Vol 14 (06) ◽  
pp. 1573-1604
Author(s):  
Alexander Dahl
Keyword(s):  
Functional Equation ◽  
Positive Integer ◽  
Dirichlet Series ◽  
Upper Bound ◽  
Dihedral Group ◽  
Central Point ◽  
Double Dirichlet Series ◽  
Dirichlet Characters ◽  
Equation Group

We study a double Dirichlet series of the form [Formula: see text], where [Formula: see text] and [Formula: see text] are quadratic Dirichlet characters with prime conductors [Formula: see text] and [Formula: see text] respectively. A functional equation group isomorphic to the dihedral group of order 6 continues the function meromorphically to [Formula: see text]. The developed theory is used to prove an upper bound for the smallest positive integer [Formula: see text] such that [Formula: see text] does not vanish. Additionally, a convexity bound at the central point is established to be [Formula: see text] and a subconvexity bound of [Formula: see text] is proven. An application of bounds at the central point to the non-vanishing problem is also discussed.

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