Zero-density estimates for automorphic <Emphasis Type="Italic">L</Emphasis>-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.     

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.     

On the central value of symmetric square L-functions

Let S _k (N, χ) be the space of cusp forms of weight k, level N and character χ. For let L(s, sym² f) be the symmetric square L-function and be the Rankin–Selberg square attached to f. For fixed k ≥ 2, N prime, and real primitive χ, asymptotic formulas for the first and second moment of the central value of L(s, sym² f) and over a basis of S _k (N, χ) are given as N → ∞. As an application it is shown that a positive proportion of the central values L(1/2, sym² f) does not vanish. The author was supported by NSERC grant 311664-05.     

Functoriality of automorphic <Emphasis Type="Italic">L</Emphasis>-functions through their zeros

Let E be a Galois extension of ℚ of degree l, not necessarily solvable. In this paper we first prove that the L-function L(s, π) attached to an automorphic cuspidal representation π of 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, π _k ), where π _j , j = 1,…, k, are automorphic cuspidal representations of , 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 , j = 1,…,k. In particular, if π is not invariant under the action of any nontrivial σ ∈ Gal _E/ℚ, then L(s, π) must equal a single L-function attached to a cuspidal representation of 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,m ₁,…,m _k are small, but are under Hypothesis H and a bound toward the Ramanujan conjecture in other cases. The first author was supported by the National Basic Research Program of China, the National Natural Science Foundation of China (Grant No. 10531060), and Ministry of Education of China (Grant No. 305009). The second author was supported by the National Security Agency (Grant No. H98230-06-1-0075). The United States Government is authorized to reproduce and distribute reprints notwithstanding any copyright notation herein     

A proof of Selberg's orthogonality for automorphic L-functions

Let π and π′ be automorphic irreducible cuspidal representations of GL_m(Q_A) and GL_m′(Q_A), respectively. Assume that π and π′ are unitary and at least one of them is self-contragredient. In this article we will give an unconditional proof of an orthogonality for π and π′, weighted by the von Mangoldt function Λ(n) and 1−n/x. We then remove the weighting factor 1−n/x and prove the Selberg orthogonality conjecture for automorphic L-functions L(s,π) and L(s,π′), unconditionally for m≤4 and m′≤4, and under the Hypothesis H of Rudnick and Sarnak [20] in other cases. This proof of Selberg's orthogonality removes such an assumption in the computation of superposition distribution of normalized nontrivial zeros of distinct automorphic L-functions by Liu and Ye [12].     

Exponential sums over primes formed with coefficients of primitive cusp forms

Let Af（n） be the coefficient of the logarithmic derivative for the Hecke L-function. In this paper we study the cancellation of the function Ay（n） twisted with an additive character e（α√n）, α 〉0, i.e. Ef（x） = Σx〈n〈2x Af（n）e（α√n）.     

L-Functions of Jacobi forms with Shimura type

For every Jacobi form of Shimura type over H × ℂ, a system of L-functions associated to it is given. These L-functions can be analytically continued to the whole complex plane and satisfy a kind of functional equation. As a consequence, Hecke’s inverse theorem on modular forms is extended to the context of Jacobi forms with Shimura type.     

On fractional moments of Dirichlet <Emphasis Type="Italic">L</Emphasis>-functions. III

In this paper, we consider upper and lower bounds of the same order with explicitly given constants for fractional moments of Dirichlet L-functions. Published in Lietuvos Matematikos Rinkinys, Vol. 47, No. 2, pp. 280–295, April–June, 2007.     

K2 of a Fermat quotient and the value of its L-function

Two linearly independent elements of k ₂ of a certain quotient of a Fermat curve is exhibited in an explicit manner. The covolume of the regulator and the value of the L-function of the curve is numerically computed, and their ratio is nearly equal to a simple rational number.     

On the 2<Emphasis Type="Italic">k</Emphasis>-th Power Mean of Inversion of <Emphasis Type="Italic">L</Emphasis>-functions with the Weight of the Gauss Sum

Abstract 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. This work is supported by the Doctorate Foundation of Xi’an Jiaotong University     

On Fractional Moments of Dirichlet <Emphasis Type="Italic">L</Emphasis>-Functions

We obtain upper and lower bounds of the same order for fractional moments of Dirichlet L-functions.Supported by a grant from the Lithuanian Science and Studies Fundation.__________Translated from Lietuvos Matematikos Rinkinys, Vol. 45, No. 2, pp. 208–228, April–June, 2005.     

Converse theorems assuming a partial euler product

Associated to a newform f(z) is a Dirichlet series L _f (s) with functional equation and Euler product. Hecke showed that if the Dirichlet series F(s) has a functional equation of a particular form, then F(s)=L _f (s) for some holomorphic newform f(z) on Γ(1). Weil extended this result to Γ₀(N) under an assumption on the twists of F(s) by Dirichlet characters. Conrey and Farmer extended Hecke’s result for certain small N, assuming that the local factors in the Euler product of F(s) were of a special form. We make the same assumption on the Euler product and describe an approach to the converse theorem using certain additional assumptions. Some of the assumptions may be related to second order modular forms. This work resulted from an REU at Bucknell University and the American Institute of Mathematics. Research supported by the American Institute of Mathematics and the National Science Foundation.     

<Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>-mixed intersection bodies

In this paper the author first introduce a new concept of L _p -dual mixed volumes of star bodies which extends the classical dual mixed volumes. Moreover, we extend the notions of L _p intersection body to L _p -mixed intersection body. Inequalities for L _p -dual mixed volumes of L _p -mixed intersection bodies are established and the results established here provide new estimates for these type of inequalities. This work was supported by the Natural Science Foundation of Zhejiang Province of China (Grant No. Y605065) and the Foundation of the Education Department of Zhejiang Province of China (Grant No. 20050392)     

On fractional moments of Dirichlet <Emphasis Type="Italic">L</Emphasis>-functions,II

We obtain upper and lower bounds for fractional moments of Dirichlet L-functions. Published in Lietuvos Matematikos Rinkinys, Vol. 46, No. 4, pp. 606–621, October–December, 2006.     

Estimates of a certain sum involving coefficients of cusp forms in weight and level aspects

We consider the sum of coefficients which are in the Dirichlet series expansion of symmetric square L-functions. In this paper, we obtain two estimates of this sum in weight and level aspects. These imply two estimates of the sum of the n ²th Fourier coefficients of cusp forms.     

Averaging Special Values of Dirichlet <Emphasis Type="Italic">L</Emphasis>-Series

In this paper we derive estimates for weighted averages of the special values of Dirichlet L-series which generalize similar estimates of David and Pappalardi [1]. The author is partially supported by NSF grant DMS-0090117. 2000 Mathematics Subject Classification: Primary–11M06; Secondary–11G05     

Real zeros of Eisenstein series and Rankin-Selberg L-functions

We prove that the Eisenstein series E(z, s) have no real zeroes for s ∈ (0, 1) when the value of the imaginary part of z is in the range $\tfrac{1}{5}$ < Im z < 4.94. For very large and very small values of the imaginary part of z, E(z, s) have real zeros in (½, 1), i.e. GRH does not hold for the Eisenstein series. Using these properties, we prove that the Rankin-Selberg L-function attached with the Ramanujan τ-function has no real zeros in the critical strip, except at the central point s = ½.     

The mean square of the DirichletL-functions in the critical strip

We study an asymptotic formula of the DirichletL-functions in the critical strip. This is an analogy of the Atkinson-type formula for DirichletL-functions. Published in Lietuvos Matematikos Rinkinys, Vol. 40, No. 2, pp. 201–213, April–June, 2000.     

<Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript> discrepancy of generalized two-dimensional Hammersley point sets

We determine the L _p discrepancy of the two-dimensional Hammersley point set in base b. These formulas show that the L _p discrepancy of the Hammersley point set is not of best possible order with respect to the general (best possible) lower bound on L _p discrepancies due to Roth and Schmidt. To overcome this disadvantage we introduce permutations in the construction of the Hammersley point set and show that there always exist permutations such that the L _p discrepancy of the generalized Hammersley point set is of best possible order. For the L ₂ discrepancy such permutations are given explicitly. F.P. is supported by the Austrian Science Foundation (FWF), Project S9609, that is part of the Austrian National Research Network “Analytic Combinatorics and Probabilistic Number Theory”.     

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 .