Modular integrals on Γo(N) and dirichlet series with functional equations

1985 ◽  
pp. 211-224 ◽  
Author(s):  
Marvin I. Knopp
Keyword(s):  
Dirichlet Series ◽  
Functional Equations ◽  
Modular Integrals

scholarly journals Weil type representations and automorphic forms

1980 ◽  
Vol 77 ◽  
pp. 145-166 ◽  
Author(s):  
Toshiaki Suzuki
Keyword(s):  
Dirichlet Series ◽  
Functional Equations ◽  
Theta Series ◽  
Type Representation ◽  
Generalized Poisson ◽  
Summation Formulae ◽  
Reciprocity Law ◽  
Weil Type ◽  
Generalized Theta Series ◽  
Poisson Summation

During 1934-1936, W. L. Ferrar investigated the relation between summation formulae and Dirichlet series with functional equations, inspired by Voronoi’s works (1904) on summation formula with respect to the numbers of divisors. In [11] A. Weil showed that the automorphic properties of theta series are expressed by generalized Poisson summation formulae with respect to the so-called Weil representation. On the other hand, T. Kubota, in his study of the reciprocity law in a number field, defined a generalized theta series and a generalized Weil type representation of SL(2, C) and obtained similar results to those of A. Weil (1970-1976, [5], [6], [7]). The methods, used by W. L. Ferrar and T. Kubota, to obtain (generalized Poisson) summation formulae depend similarly on functional equations of Dirichlet series (attached to the generalized theta series).

scholarly journals Subconvexity for a double Dirichlet series

2010 ◽  
Vol 147 (2) ◽  
pp. 355-374 ◽  
Author(s):  
Valentin Blomer
Keyword(s):  
Dirichlet Series ◽  
Functional Equations ◽  
Mean Square ◽  
Double Dirichlet Series ◽  
Image Position

AbstractFor two real characters ψ,ψ′ of conductor dividing 8 define where $\chi _d = (\frac {d}{.})$ and the subscript 2 denotes the fact that the Euler factor at 2 has been removed. These double Dirichlet series can be extended to $\Bbb {C}^2$ possessing a group of functional equations isomorphic to D12. The convexity bound for Z(s,w;ψ,ψ′) is |sw(s+w)|1/4+ε for ℜs=ℜw=1/2. It is proved that Moreover, the following mean square Lindelöf-type bound holds: for any Y1,Y2≥1.

Weyl Group Multiple Dirichlet Series

2011 ◽  
Author(s):  
Ben Brubaker ◽  
Daniel Bump ◽  
Solomon Friedberg
Keyword(s):  
Zeta Function ◽  
Dirichlet Series ◽  
Weyl Group ◽  
Riemann Zeta Function ◽  
Functional Equations ◽  
Lattice Points ◽  
Baxter Equation ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
Multiple Dirichlet Series

Weyl group multiple Dirichlet series are generalizations of the Riemann zeta function. Like the Riemann zeta function, they are Dirichlet series with analytic continuation and functional equations, having applications to analytic number theory. By contrast, these Weyl group multiple Dirichlet series may be functions of several complex variables and their groups of functional equations may be arbitrary finite Weyl groups. Furthermore, their coefficients are multiplicative up to roots of unity, generalizing the notion of Euler products. This book proves foundational results about these series and develops their combinatorics. These interesting functions may be described as Whittaker coefficients of Eisenstein series on metaplectic groups, but this characterization doesn't readily lead to an explicit description of the coefficients. The coefficients may be expressed as sums over Kashiwara's crystals, which are combinatorial analogs of characters of irreducible representations of Lie groups. For Cartan Type A, there are two distinguished descriptions, and if these are known to be equal, the analytic properties of the Dirichlet series follow. Proving the equality of the two combinatorial definitions of the Weyl group multiple Dirichlet series requires the comparison of two sums of products of Gauss sums over lattice points in polytopes. Through a series of surprising combinatorial reductions, this is accomplished. The book includes expository material about crystals, deformations of the Weyl character formula, and the Yang–Baxter equation.

scholarly journals Functional Equations of Dirichlet Series Derived from Non-Analytic Automorphic Forms of a Certain Type

1975 ◽  
Vol 18 (1) ◽  
pp. 87-94 ◽  
Author(s):  
V. Venugopal Rao
Keyword(s):  
Real Number ◽  
Dirichlet Series ◽  
Complex Number ◽  
Functional Equations ◽  
Fourier Expansion ◽  
Positive Real Number ◽  
Automorphic Forms ◽  
Positive Real ◽  
Real Value ◽  
Complex Valued

Let f(τ) be a complex valued function, defined and analytic in the upper half of the complex τ plane (τ = x+iy, y > 0), such that f(τ + λ)= f(τ) where λ is a positive real number and f(—1/τ) = γ(—iτ)kf(τ), k being a complex number. The function (—iτ)k is defined as exp(k log(—iτ) where log(—iτ) has the real value when —iτ is positive. Every such function is said to have signature (λ, k, γ) in the sense of E. Hecke [1] and has a Fourier expansion of the type f(τ) = a0 + σ an exp(2πin/λ), (n = 1,2,…), if we further assume that f(τ) = O(|y|-c) as y tends to zero uniformly for all x, c being a positive number.

Dirichlet series with functional equations and arithmetical identities

2015 ◽  
Vol 41 (1-3) ◽  
pp. 115-146
Author(s):  
Sister Ann M. Heath
Keyword(s):  
Dirichlet Series ◽  
Functional Equations
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