The Generalized Prime Number Theorem for Automorphic L-Functions

Let π and π' be automorphic irreducible cuspidal representations of GLm (QA) and GLm', (QA), respectively, and L(s,π×(～π)') be the Rankin-Selberg L-function attached to π and π'. Without assuming the Generalized Ramanujan Conjecture (GRC), the author gives the generalized prime number theorem for L(s, π×(～π)') when π(=)π'. The result generalizes the corresponding result of Liu and Ye in 2007.     

A Limit Theorem for Dirichlet L-Functions

We prove a limit theorem on the weak convergence of probability measures in the space of continuous functions for Dirichlet L-functions. The result generalizes a similar theorem for the Riemann zeta-function.     

Simultaneous Nonvanishing of Twists of Automorphic L-Functions

Given three distinct primitive complex characters ₁,₂,₃ satisfying some technical conditions, we prove that the triple product of twisted L-functions L(f·₁,1/2) L(f·₂,1/2) L(f·₃,1/2) does not vanish for a positive proportion of weight 2 primitive forms for ₀(q), when q goes to infinity through the set of prime numbers. This result, together with some variants, implies the existence of quotients of J ₀(q) of large dimension satisfying the Birch–Swinnerton-Dyer conjecture over cyclic number fields of degree less than 5.P.M. is partially supported by NSF Grant DMS-97-2992 and by the Ellentuck fund (by grants to the Institute for Advanced Study) and by the Institut Universitaire de France.     

On the Equivalence and Generalized of Weyl Theorem Weyl Theorem

We know that an operator T acting on a Banach space satisfying generalized Weyl＇s theorem also satisfies Weyl＇s theorem. Conversely we show that if all isolated eigenvalues of T are poles of its resolvent and if T satisfies Weyl＇s theorem, then it also satisfies generalized Weyl＇s theorem. We give also a sinlilar result for the equivalence of a-Weyl＇s theorem and generalized a-Weyl＇s theorem. Using these results, we study the case of polaroid operators, and in particular paranormal operators.     

Extremal Values of L-Functions Associated with Newforms in the Half-Plane of Absolute Convergence

We prove estimates for extremal values of L-functions associated with newforms f in the half-plane of absolute convergence of their Dirichlet series expansion. The proof is based on an effective version of Kronecker's approximation theorem and estimates for the Fourier coefficients of the newform f.     

A-Browder＇s Theorem and Generalized a-Weyl＇s Theorem

Two variants of the essential approximate point spectrum are discussed. We find for example that if one of them coincides with the left Drazin spectrum then the generalized a-Weyl＇s theorem holds, and conversely for a-isoloid operators. We also study the generalized a-Weyl＇s theorem for Class A operators.     

关于抽象素数定理的形式(英文)

In this paper we prove a zero-free region for L-functions LG（z,Х）. As an application, an abstract prime number theorem with sharp error-term for formations is established.     

Determining cusp forms by central values of Rankin-Selberg L-functions

Let g be a fixed normalized Hecke-Maass cusp form for SL(2,Z) associated to the Laplace eigenvalue . We show that g is uniquely determined by the central values of the family for k sufficiently large, where Hk(1) denotes a Hecke basis of the space of holomorphic cusp forms for SL(2,Z).     

On a sum involving Fourier coefficients of cusp forms

We improve the existing upper bound for the quantity |∑ _n⩽x a(n ²)|, where a(n ²) is the n ²th Hecke eigenvalue of a normalized holomorphic cusp form (Hecke eigenform) of the full modular group SL(2, ℤ), whenever the weight of the original holomorphic cusp form (Hecke eigenform) lies in a certain range. Published in Lietuvos Matematikos Rinkinys, Vol. 46, No. 4, pp. 565–583, October–December, 2006.     

Prime number theorems for Rankin-Selberg L-functions over number fields

In this paper we define a Rankin-Selberg L-function attached to automorphic cuspidal represen-tations of GLm(AE) × GLm (AF ) over cyclic algebraic number fields E and F which are invariant under the Galois action,by exploiting a result proved by Arthur and Clozel,and prove a prime number theorem for this L-function.     

Eisenstein Cohomology and the Construction of p-Adic Analytic L-Functions

Let be a cuspidal automorphic representation of GL₃( ), unramified at pand of cohomological type at infinity. We construct p-adic L-functions, which interpolate the critical values of L(,s) and which satisfy a logarithmic growth condition. We obtain these functions as p-adic Mellin transforms of certain distributions on _p ^* having values in some fixed number field and which are of moderate growth. In the p-ordinary case we obtain the bound |(U)| _p |_Haar(U)| _p for open subsets U _p ^*, where _Haardenotes the invariant distribution on _p ^*.     

Stickelberger Elements for Cyclic Extensions and the Order of Vanishing of Abelian L-Functions at s=0

We study the Stickelberger element of a cyclic extension of global fields of prime power degree. Assuming that S contains an almost splitting place, we show that the Stickelberger element is contained in a power of the relative augmentation ideal whose exponent is at least as large as Gross's prediction. This generalizes the work of Tate (see Section 4) on a refinement of Gross's conjecture in the cyclic case. We also present an example for which Tate's prediction does not hold.     

Extreme Values of Artin L-Functions and Class Numbers

Assuming the generalized Riemann hypothesis (GRH) and Artin conjecture for Artin L-functions, we prove that there exists a totally real number field of any fixed degree (>1) with an arbitrarily large discriminant whose normal closure has the full symmetric group as Galois group and whose class number is essentially as large as possible. One ingredient is an unconditional construction of totally real fields with small regulators. Another is the existence of Artin L-functions with large special values. Assuming the GRH and Artin conjecture it is shown that there exist an Artin L-functions with arbitrarily large conductor whose value at s=1 is extremal and whose associated Galois representation has a fixed image, which is an arbitrary nontrivial finite irreducible subgroup of GL(n, ) with property Gal _T .     

Algebraic independence of the values ofE-functions at singular points and Siegel’s conjecture

We prove a generalization of Shidlovskii’s theorem on the algebraic independence of the values ofE-functions satisfying a system of linear differential equations that is well known in the theory of transcendental numbers. We consider the case in which the values ofE-functions are taken at singular points of these systems. Using the obtained results, we prove Siegel’s conjecture that, for the case of first-order differential equations, anyE-function satisfying a linear differential equation is representable as a polynomial in hypergeometricE-functions. Translated fromMatematicheskie Zametki, Vol. 67, No. 2, pp. 174–190, February, 2000.     

On the Number of Birch Partitions

Birch and Tverberg partitions are closely related concepts from discrete geometry. We show two properties for the number of Birch partitions: Evenness and a lower bound. This implies the first nontrivial lower bound for the number of Tverberg partitions that holds for arbitrary q, where q is the number of partition blocks. The proofs are based on direct arguments and do not use the equivariant method from topological combinatorics.     

Central Value Of Automorphic <Emphasis Type="Italic">L</Emphasis>-Functions

We prove a generalization to the totally real field case of the Waldspurger’s formula relating the Fourier coefficient of a half integral weight form and the central value of the L-function of an integral weight form. Our proof is based on a new interpretation of Waldspurger’s formula as a combination of two ingredients – an equality between global distributions, and a dichotomy result for theta correspondence. As applications we generalize the Kohnen–Zagier formula for holomorphic forms and prove the equivalence of the Ramanujan conjecture for half integral weight forms and a case of the Lindel?f hypothesis for integral weight forms. We also study the Kohnen space in the adelic setting. The first author was partially supported by NSF grant DMS-0070762. The second author was partially supported by NSF grant DMS-0355285. Received: July 2005 Accepted: August 2005     

Moments of Generalized Quadratic Gauss Sums Weighted by L-Functions

The main purpose of this paper is using estimates for character sums and analytic methods to study the second, fourth, and sixth order moments of generalized quadratic Gauss sums weighted by L-functions. Three asymptotic formulae are obtained.     

The Exterior Cube L-Function for GL(6)

We construct a Rankin Selberg integral to represent the exterior cube L function L(,³,s) of an automorphic cuspidal module of GL₆( _F ) (where F is a number field). We determine the poles of this L function and find period conditions for the special value L(,³,1/2). We use the Siegal Weil formula. We also state an analogue of the Gross–Prasad conjecture concerning a criterion for the nonvanishing of L(,³,1/2).     

An L-Function of Degree 27 for Spin9

This paper studies a Rankin-Selberg integral for a degree 27 L-function on Spin(9). It makes use of an Eisenstein series on the exceptional group F ₄.