On the Newton polygons of twisted L-functions of binomials

Let χ be an order c multiplicative character of a finite field and $f(x)=x^d+\lambda x^e$ a binomial with $(d,e)=1$. We study the twisted classical and T-adic Newton polygons of f. When $p>(d-e)(2d-1)$, we give a lower bound of Newton polygons and show that they coincide if p does not divide a certain integral constant depending on $p\mathrm{mod}cd$.We conjecture that this condition holds if p is large enough with respect to $c,d$ by combining all known results and the conjecture given by Zhang-Niu. As an example, we show that it holds for $e=d-1$.     

Integral representation and estimation of harmonic functions in the quaternionic half space

In this article, we discuss the integral representation of quaternionic harmonic functions in the half space with the general boundary condition. Next, we derive a lower bound from an upper one for quaternionic harmonic functions. These results generalize some of the classic results from the case of plane to the case of noncommutative quaterninionic half space. Copyright © 2017 John Wiley & Sons, Ltd.     

Weighted scattering matrices of regular coverings of graphs

We give a decomposition formula for the determinant det(I ? U(λ)) of the weighted bond scattering matrix U(λ) of a regular covering of G. Furthermore, we define an L-function of G, and give a determinant expression of it. As a corollary, we express some determinant of the weighted bond scattering matrix of a regular covering of G by means of its L-functions.     

On an open problem of Sankaranarayanan

Let λ(n) be the n-th normalized Fourier coeficient of a holomorphic Hecke eigenform f(z) ∈ Sk(Γ) for the full modular group.In one of his papers,Sankaranarayanan mentioned that it is an open problem to give a non-trivial estimate for the sum of λ(n) over cubes,i.e.n x λ(n3).In this paper,we are able to use the analytic properties of symmetric power L-functions to solve his problem.More precisely,we prove that Σn≤zλ(n3) x(3/4 +ε),Σn≤zλ(n4) x(97+ε).     

Applications of an Explicit Formula for the Generalized Euler Numbers

The authors establish an explicit formula for the generalized Euler NumbersE2n^（x）, and obtain some identities and congruences involving the higher＇order Euler numbers, Stirling numbers, the central factorial numbers and the values of the Riemann zeta-function.     

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 factoriz...     

Several Expressions of Dirichlet L-function at Positive Integers

In this paper, by making use of Abel's theorem on power series, the reflection formula and the function equation for Hurwitz zeta function, we establish several expressions of Dirichlet L-function at positive integers by means of some finite sums of different types. Some special cases as well as immediate consequences of the results presented here are also considered.     

Discrete Universality of L-Functions for New Forms

A Voronin-type discrete universality theorem for the L-functions of new forms is proved.     

关于Siegel-Tatuzawa定理

设0<ε<1/（6log10）,x为模k的实原特征且k大于e1/ε.则除去至多一个可能的例外特征外,均有这改进了Hoffstein等人的结果.