Higher moments of certain <Emphasis Type="Italic">L</Emphasis>-functions on the critical line

We study the sixth-power moments of certain L-functions belonging to a sub-class of the Selberg’s class on the critical line and, using this, we conclude an upper bound for the fourth-power moments of certain L-functions related to GL ₃ on the critical line. This is an analogue of the upper bound for the twelfth-power moment of the Riemann zeta-function on the critical line. Published in Lietuvos Matematikos Rinkinys, Vol. 47, No. 3, pp. 341–380, July–September, 2007.     

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.     

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.     

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.     

Computation of the Rankin–Selberg <Emphasis Type="Italic">L</Emphasis>-functions of Maass wave forms

This note outlines a method for numerically computing the Rankin–Selberg convolutions of Maass wave forms L-functions and reports on the computation of zeros of some of them. Bibliography: 11 titles.     

On <Emphasis Type="Italic">g</Emphasis>-functions for Laguerre function expansions of Hermite type

We examine weighted L ^p boundedness of g-functions based on semigroups related to multi-dimensional Laguerre function expansions of Hermite type. A technique of vector-valued Calderón–Zygmund operators is used.     

On special values of certain Dirichlet L-functions

Let r _k (n) denote the number of ways n can be expressed as a sum of k squares. Recently, S. Cooper (Ramanujan J. 6:469–490, [2002]), conjectured a formula for r ₉(t), t≡5 (mod 8), r ₁₁(t), t≡7 (mod 8), where t is a square-free positive integer. In this note we observe that these conjectures follow from the works of Lomadze (Akad. Nauk Gruz. Tr. Tbil. Mat. Inst. Razmadze 17:281–314, [1949]; Acta Arith. 68(3):245–253, [1994]). Further we express r ₉(t), r ₁₁(t) in terms of certain special values of Dirichlet L-functions. Combining these two results we get expressions for these special values of Dirichlet L-functions involving Jacobi symbols.      

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.     

Pairings of automorphic distributions

We present a pairing of automorphic distributions that applies in situations where a Lie group acts with an open orbit on a product of generalized flag varieties. The pairing gives meaning to an integral of products of automorphic distributions on these varieties. This generalizes classical integral representations or “Rankin–Selberg integrals” of L-functions, and gives new constructions and analytic continuations of automorphic L-functions.     

Nonvanishing of Hecke <Emphasis Type="Italic">L</Emphasis>-functions and the Bloch–Kato conjecture

In this paper we study the central values of L-functions associated to a large class of algebraic Hecke characters of imaginary quadratic fields. When these central values are nonzero, the Bloch–Kato conjecture predicts an exact formula for the algebraic parts of the central values in terms of periods and arithmetic data, most notably the Selmer groups corresponding to the Hecke characters. We investigate the nonvanishing of these central values, and prove the p-part of the Bloch–Kato conjecture in these cases for primes p which split in K.     

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     

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     

Values of <Emphasis Type="Italic">L</Emphasis>-functions at integers outside the critical strip

We incorporate the non-critical values of L-functions of cusp forms into a cohomological set-up analogous to the one of Eichler, Manin and Shimura. We use the 1-cocycles we associate in this way to non-critical values to prove an expression for such values which is similar in structure to Manin’s formula for the critical value of the L-function of a weight 2 cusp form. YoungJu Choie is partially supported by KOSEF R01-2003-00011596-0 and by ITRC Research Fund. N. Diamantis is partially supported by EPSRC grant EP/D032350/1.     

Value-distribution of twisted automorphic <Emphasis Type="Italic">L</Emphasis>-functions

In this paper, we prove a limit theorem for twisted with character automorphic L-functions with an increasing modulus of the character.     

L-functions of second-order cusp forms

We discuss equivalent definitions of holomorphic second-order cusp forms and prove bounds on their Fourier coefficients. We also introduce their associated L-functions, prove functional equations for twisted versions of these L-functions and establish a criterion for a Dirichlet series to originate from a second order form. In the last section we investigate the effect of adding an assumption of periodicity to this criterion. 2000 Mathematics Subject Classification Primary—11F12, 11F66 G. Mason: Research supported in part by NSF Grant DMS 0245225. C. O’Sullivan: Research supported in part by PSC CUNY Research Award No. 65453-00 34.     

On zeros of Hecke <Emphasis Type="Italic">L</Emphasis>-functions and their linear combinations on the critical line

In this paper two theorems were obtained. In the first theorem it is proved that a positive proportion of non-trivial zeros lie on the critical line for L-functions attached to automorphic cusp forms for congruence-subgroups. Therefore, the class of functions satisfying a variant of Selberg’s theorem was extended. In the second theorem a new lower bound was obtained for the number of zeros of linear combinations of Hecke L-functions on the intervals of the critical line. This theorem essentially improves the previously known S.A. Gritsenko’s result of 1997.     

Cohomology of quaternionic Shimura varieties at parahoric levels

 Some semi-simple L-functions which are associated with the cohomology of a quaternionic Shimura variety are compared with semi-simple automorphic L-functions. Assuming a certain purity condition this yields a similar result for the usual L-functions. The main theorem of the present paper extends previous results of the author to a more general case. Received: 19 July 2000     

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].     

Computing the values of <Emphasis Type="Italic">L</Emphasis>-functions,and the chebyshev polynomials

The Chebyshev polynomials and Chebyshev’s economization method are applied to speed up the computation of the values of L-functions. Bibliography: 3 titles.