Euler Product Expressions of Absolute Tensor Products of Dirichlet L-Functions

In this paper, we calculate the absolute tensor square of the Dirichlet L-functions and show that it is expressed as an Euler product over pairs of primes. The method is to construct an equation to link primes to a series which has the factors of the absolute tensor product of the Dirichlet L-functions. This study is a generalization of Akatsuka’s theorem on the Riemann zeta function, and gives a proof of Kurokawa’s prediction proposed in 1992.

A Zero Density Estimate for Automorphic L-functions

1 Introduction Let f be a holomorphic cusp form for the group F = SL2（Z） of even integral weight k, with Fourier coefficients af （n）.

UNIQUENESS THEOREMS OF L-FUNCTIONS IN THE EXTENDED SELBERG CLASS

We establish uniqueness theorems of L-functions in the extended Selberg class,which show how an L-function and a meromorphic function are uniquely determined by their shared values in two finite sets.This can be seen as a new solution of a problem proposed by Gross.

Grand Zero Density Estimates of L-functions Associated with Cusp Forms

In this paper a zero-density estimate of the large sieve type is given for the automorphic L-function L f (s,χ),where f is a holomorphic cusp form and χ a Dirichlet character of mod q.

Value Distribution of L-Functions with Rational Moving Targets

We prove some value-distribution results for a class of L-functions with rational moving targets. The class contains Selberg class, as well as the Riemann-zeta function.

A New Zero-density Result of L-functions Attached to Maass Forms

Let f denote a normalized Maass cusp form for SL（2, Z）, which is an eigenfunction of all the Hecke operators T（n） as well as the reflection operator T-1： z →z. We obtain a zero-density result of the L-function attached to f near σ = 1. This improves substantially the previous results in this direction.

On the 2k-th Power Mean of Inversion of L-functions with the Weight of the Gauss Sum

The main purpose of this paper is to use the estimate for character sums and the method of trigonometric sums to study the 2k-th power mean of the inversion of Dirichlet L-functions with the weight of the Gauss sums, and give a sharper asymptotic formula.

Periods of automorphic forms,poles of L-functions and functorial lifting

In this paper,we survey our work on period,poles or special values of certain automorphic Lfunctions and their relations to certain types of Langlands functorial transfers.We outline proofs for some results and leave others as conjectures.

ON THE FOURTH POWER MEAN OF DIRICHLET L-FUNCTIONS

I. INTRODUCTIONFor an integer q】2, 1et x denote a typical Dirichlet character mod q, x0 the principal character, and L（s, x） the corresponding Dirichlet L-function. The main purpose of this note is to give a more accurate asymptotic formula for the fourth power

Hybrid subconvexity bounds for twisted L-functions on GL(3)

Let q be a large prime, and χ the quadratic character modulo q. Let Φ be a self-dual Hecke-Maass cusp form for SL(3, Z), and uj a Hecke-Maass cusp form for Γ0(q) ■ SL(2, Z) with spectral parameter tj. We prove, for the first time, some hybrid subconvexity bounds for the twisted L-functions on GL(3), such as L(1/2, Φ× uj ×χ) ■Φ,ε(q(1 + |tj |))3/2-θ+ε and L(1/2 + it, Φ×χ) ■Φ,ε(q(1 + |t|))3/4-θ/2+ε for any ε > 0, where θ = 1/23 is admissible. The proofs depend on the first moment of a family of L-functions in short intervals. In order to bound this moment, we first use the approximate functional equations, the Kuznetsov formula, and the Voronoi formula to transform it to a complicated summation, and then we apply different methods to estimate it, which give us strong bounds in different aspects. We also use the stationary phase method and the large sieve inequalities.

Some Notes on Identities for Dirichlet L-functions

Let q ≥ 3 be an integer, and χ be a Dirichlet character modulo q, L（s, χ） denote the Dirichlet L-function corresponding to χ. In this paper, we show some early information on the mean value of the Dirichlet L-functions and give some new identities forwhere a = 2, 3, or 4. Then we give general identities for the case that the integer a divides q - 1. Keywords Dirichlet L-functions, Dedekind sum, trigonometric formula, MSbius inversion formula

Selberg's Normal Density Theorem for Automorphic L-Functions for GL_m

Let π be an irreducible unitary cuspidal representation of GLm（AQ） with m ≥ 2, and L（s, Tr） the L-function attached to π. Under the Generalized Riemann Hypothesis for L（s,π）, we estimate the normal density of primes in short intervals for the automorphic L-function L（s, π）. Our result generalizes the corresponding theorem of Selberg for the Riemann zeta-function.

The First Power Mean of the Inversion of L-Functions Weighted by Quadratic Gauss Sums

The main purpose of this paper is to use estimates for character sums and analytic methods to study the first power mean of the inversion of Dirichlet L-functions with the weight of general quadratic Gauss sums,and three asymptotic formulae are obtained.

The second moment of GL(3)×GL(2)L-functions at special points from GL(3)forms

For a fixed even SL(2,Z)Hecke-Maass form f,we get an estimate for the second moment of L(s,φj×f)at special points,whereφj runs over an orthogonal basis of Hecke-Maass cusp forms for SL3(Z).

Correlation of zeros of automorphic L-functions

We compute the n-level correlation of normalized nontrivial zeros of a product of L-functions:L(s,π1)···L(s,πk), where πj, j=1,...,k, are automorphic cuspidal representations of GLmj(QA). Here the sizes of the groups GLmj(QA) are not necessarily the same. When these L(s,πj) are distinct, we prove that their nontrivial zeros are uncorrelated, as predicted by random matrix theory and verified numerically. When L(s,πj) are not necessarily distinct, our results will lead to a proof that the n-level correlation of normalized nontrivial zeros of the product L-function follows the superposition of Gaussian Unitary Ensemble (GUE) models of individual L-functions and products of lower rank GUEs. The results are unconditional when m1,...,mk 4,but are under Hypothesis H in other cases.

Functoriality of automorphic L-functions through their zeros

Let E be a Galois extension of Q of degree , not necessarily solvable. In this paper we first prove that the L-function L(s,π) attached to an automorphic cuspidal representation π of GLm(EA) cannot be factored nontrivially into a product of L-functions over E. Next, we compare the n-level correlation of normalized nontrivial zeros of L(s,π1)···L(s,πk), where πj, j = 1,...,k, are automorphic cuspidal representations of GLmj(QA), with that of L(s,π). We prove a necessary condition for L(s,π) having a factorization into a product of L-functions attached to automorphic cuspidal representations of specific GLmj(QA), j = 1,...,k. In particular, if π is not invariant under the action of any nontrivial σ∈ GalE/Q, then L(s,π) must equal a single L-function attached to a cuspidal representation of GLm (QA) and π has an automorphic induction, provided L(s,π) can factored into a product of L-functions over Q. As E is not assumed to be solvable over Q, our results are beyond the scope of the current theory of base change and automorphic induction. Our results are unconditional when m,m1,...,mk are small, but are under Hypothesis H and a bound toward the Ramanujan conjecture in other cases.

Zero density of L-functions related to Maass forms

Let f（z） be a Hecke-Maass cusp form for SL2（Z）, and let L（s, f） be the corresponding automorphic L-function associated to f. For sufficiently large T, let N（σ, T） be the number of zeros p =β ＋ iγ of L（s, f） with ｜γ｜ ≤T, β〉 σ, the zeros being counted according to multiplicity. In this paper, we get that for 3/4≤ σ≤ 1 - ε, there exists a constant C = C（ε） such that N（σ, T） 〈〈 T2（1-σ）/σ（log T）c, which improves the previous results.

A NEW MEAN VALUE FORMULA OF DIRICHLET'S L-FUNCTIONS

By using the estimates of the character sums and the Bombieri-Vinogradov’s theorem, anew mean value formula of Dirichlet’s L--functions is derived.

Zero-density Estimates for Automorphic L-functions

In this paper we prove zero-density estimates of the large sieve type for the automorphic L-function L（s, f × χ）, where f is a holomorphic cusp form and χ（mod q） is a primitive character.

Double square moments and subconvexity bounds for Rankin-Selberg L-functions of holomorphic cusp forms

Let f and g be holomorphic cusp forms of weights k1 and k2 for the congruence subgroups TO(N1)and Γ0(N2),respectively.In this paper the square moment of the Rankin-Selberg L-function for f and g in the aspect of both weights in short intervals is bounded,when k1^ε <<k^2<<k1^1-ε.These bounds are the mean Lindelof hypothesis in one case and subconvexity bounds on average in other cases.These square moment estimates also imply subconvexity bounds for individual L(1/2+it,f×g) for all g when f is chosen outside a small exceptional set.In the best case scenario the subconvexity bound obtained reaches the Weyl-type bound proved by Lau et al.(2006) in both the k1 and k2 aspects.