Dirichlet series in two variables

Peter, Manfred

doi:10.1515/crll.2000.039

Cited by 6 publications

(19 citation statements)

References 7 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Moreover, from Lemmas 3.3, 3.4 and the convergence of the zeta functions, we can show that Z(n, w) and Z * (n, w) converge absolutely in the domain {w ∈ C | Re(w) > m}. The Dirichlet series Z(n, w) coincides with the series L(w, n; 1, A) studied in [5], where the following lemma is proved.…”

Section: T Uenomentioning

confidence: 58%

“…(1) Lemma 3.4 shows that the zeta functions for φ 1 andφ 1 coincide up to an elementary factor with the Dirichlet seriesτ studied in [5], where A is assumed to be positive definite.…”

Section: T Uenomentioning

confidence: 98%

“…In the case m = 2, (p, q) = (0, 2) and [5] under the assumption that A be positive definite. Moreover he posed a conjecture that the zeta functions are related to modular forms.…”

Section: T Uenomentioning

confidence: 99%

See 2 more Smart Citations

Modular forms arising from zeta functions in two variables attached to prehomogeneous vector spaces related to quadratic forms

Ueno

2004

Nagoya Mathematical Journal

View full text Add to dashboard Cite

show abstract

Section: T Uenomentioning

confidence: 58%

“…(1) Lemma 3.4 shows that the zeta functions for φ 1 andφ 1 coincide up to an elementary factor with the Dirichlet seriesτ studied in [5], where A is assumed to be positive definite.…”

Section: T Uenomentioning

confidence: 98%

See 1 more Smart Citation

Modular forms arising from zeta functions in two variables attached to prehomogeneous vector spaces related to quadratic forms

Ueno

2004

Nagoya Mathematical Journal

View full text Add to dashboard Cite

show abstract

“…It doesn't seem that the functional equation ( are absolutely convergent for Re(w) > 1. From [16] or [14] it follows that D and E l , l = 0, 1, have a meromorphic continuation to C and satisfy the functional equation…”

Section: The Voronoi Sum Formulamentioning

confidence: 99%

“…Shintani [16] proved their meromorphic continuation to C 2 and the functional equation (2.5) below (see also [14] for a generalization). Fix 1/2 ≤ σ 0 ≤ 1 (s 0 = 1 if j = 1) and define…”

Section: Introductionmentioning

confidence: 99%

Mean values of Dirichlet L-series

Peter

2000

Mathematische Annalen

Self Cite

View full text Add to dashboard Cite

show abstract

Converse theorems for automorphic distributions and Maass forms of level N

et al. 2019

View full text Add to dashboard Cite

We investigate the relations for L-functions satisfying certain functional equation, summationa formulas of Voronoi-Ferrar type and Maass forms of integral and halfintegral weight. Summation formulas of Voronoi-Ferrar type can be viewed as an automorphic property of distribution vectors of non-unitary principal series representations of the double covering group of SL(2). Our goal is converse theorems for automorphic distributions and Maass forms of level N characterizing them by analytic properties of the associated L-functions. As an application of our converse theorems, we construct Maass forms from the two-variable zeta functions related to quadratic forms defines an automorphic distribution for the group ∆(N ) = ñ(1),ñ(N ) , and hence its Poisson transform, which is given by F α (z) in (III), is a Maass form automorphic for ∆(N ). The transformation formula (0.3) is derived from the comparison of the Poisson transforms of both sides of the summation formula.The group ∆(N ) is a subgroup of the lift Γ 0 (N ) of Γ 0 (N ) to G. In §4 we prove a converse theorem that gives a condition for the automorphic distribution for ∆(N ) associated with the Ferrar-Suzuki summation formula to be automorphic for the larger group Γ 0 (N ) in terms of twists of the corresponding L-functions by Dirichlet characters (Theorem 4.2). By the Poisson transformation, this yields immediately a converse theorem for Maass forms for Γ 0 (N ) (Theorem 4.3), which is an analogue of the converse theorem of Weil [49] for holomorphic modular forms of integral weight (in the case of even ℓ) and its generalization by Shimura [40, Section 5] to holomorphic modular forms of half-integral weight (in the case of odd ℓ). A merit of our approach is to keep calculations involving the Whittaker function to a minimum. The information on the Whittaker function we need is only the fact that the Whittaker function appears as the Fourier transform of the Poisson kernel (see (2.14)). We say now a few words about the assumptions in the converse theorem. In our converse theorem some analytic properties (functional equations, location of poles and so on) are assumed for the L-functions twisted by Dirichlet characters of prime modulus including the principal characters. In the original converse theorem of Weil for holomorphic modular forms ([49]), the assumptions are imposed for the L-functions twisted only by primitive Dirichlet characters of prime modulus. As pointed out in Gelbart-Miller [9, §3.4], it has been considered difficult to transfer the argument of Weil to the Maass form case. In [2] and [3], Diamantis and Goldfeld proved a converse theorem for double Dirichlet series by considering the twists of Dirichlet series by Dirichlet characters including imprimitive characters (the principal characters), namely by the method originally found by Razar [30] for holomorphic modular forms. We follow the approach of Razar and Diamantis-Goldfeld. In a paper [25] that appeared very recently, Nuerurer and Oliver succeeded in modifying the argument of Weil to get a c...

show abstract

Dirichlet series in two variables

Cited by 6 publications

References 7 publications

Modular forms arising from zeta functions in two variables attached to prehomogeneous vector spaces related to quadratic forms

Modular forms arising from zeta functions in two variables attached to prehomogeneous vector spaces related to quadratic forms

Mean values of Dirichlet L-series

Converse theorems for automorphic distributions and Maass forms of level N

Contact Info

Product

Resources

About