Le critère de Li et l'hypothèse de Riemann pour la classe de Selberg

We generalise Li's criterion, already known for the Riemann zeta function, to a large class of Dirichlet series. We give first an explicit formula for the coefficients , for all positive integers n and ρ runs over all the non-trivial zeros of a function F in this class. To do so, we use the Weil Explicit Formula.     

The Li criterion and the Riemann hypothesis for the Selberg class II

In this paper, we prove an explicit asymptotic formula for the arithmetic formula of the Li coefficients established in Omar and Mazhouda (2007) [10] and Omar and Mazhouda (2010)[11]. Actually, for any function F(s) in the Selberg class S, we have

     

Théorie du potentiel pour des opérateurs elliptiques non linéaires du second ordre à coefficients discontinus

Let a open subset of ⁿ , n3, and an open. Existence and unicity are proved for the Dirichlet problem

Computing the Ramanujan tau function

We show that the Ramanujan tau function τ(n) can be computed by a randomized algorithm that runs in time for every O( ) assuming the Generalized Riemann Hypothesis. The same method also yields a deterministic algorithm that runs in time O( ) (without any assumptions) for every ε > 0 to compute τ(n). Previous algorithms to compute τ(n) require Ω(n) time. Research supported in part by NSF grant CCR-9988202 2000 Mathematics Subject Classification Primary—11-04; Secondary—11Y55, 11F11     

Algebraic independence of logarithmic singularities of some complex functions

We show that algebraic independence of some complex functions of one variable over regular functions implies their algebraic independence over a larger ring, containing complex powers of regular functions. Based on this we obtain a generalization of a special case of the theorem of Kaczorowski and Perelli on functional independence of logarithms of functions in the Selberg class. As an application we state a new result on oscillations of arithmetical functions.     

A mean value theorem on Dirichlet series

This paper deals with mean-value for the square of certain functionF(s) which has some characteristic properties of the Riemann zeta-function and its powers.Work supported by the University of the Basque Country.     

Riemann 映射与单连通区域分类

设G为复平面上一个单连通区域及φ为G的Riemann 映射. 本文通过φ是否属于G上多项式在不同拓扑下的闭包的情况对G进行分类. 特别地, 我们对已知的几类单连通给出了刻画.     

All but Finitely Many Non-Trivial Zeros of the Approximations of the Epstein Zeta Function are Simple and on the Critical Line

The Chowla–Selberg formula is applied in approximatinga given Epstein zeta function. Partial sums of the series derivefrom the Chowla–Selberg formula, and although these partialsums satisfy a functional equation, as does an Epstein zetafunction, they do not possess an Euler product. What we callpartial sums throughout this paper may be considered as specialcases concerning a more general function satisfying a functionalequation only. In this article we study the distribution ofzeros of the function. We show that in any strip containingthe critical line, all but finitely many zeros of the functionare simple and on the critical line. For many Epstein zeta functionswe show that all but finitely many non-trivial zeros of partialsums in the Chowla–Selberg formula are simple and on thecritical line. 2000 Mathematics Subject Classification 11M26.     

On the summation of Schlömilch's series

ABSTRACT

Schlömilch's series is named after the German mathematician Oscar Xavier Schlömilch, who derived it in 1857 as a Fourier series type expansion in terms of the Bessel function of the first kind. However, except for Bessel functions, here we consider an expansion in terms of Struve functions or Bessel and Struve integrals as well. The method for obtaining a sum of Schlömilch's series in terms of the Bessel or Struve functions is based on the summation of trigonometric series, which can be represented in terms of the Riemann zeta and related functions of reciprocal powers and in certain cases can be brought in the closed form, meaning that the infinite series are represented by finite sums. By using Krylov's method we obtain the convergence acceleration of the trigonometric series.     

The evaluation of character Euler double sums

Euler considered sums of the form

Fractional Moments and Zeros of ζ(s) on the Critical Line

A connection is established between the fractional moments of the Riemann zeta-function and the number of its zeros on the critical line.     

Hardy's Inequality for Dirichlet Forms

We prove an abstract form of Hardy's L² inequality, in which the Dirichlet integral is replaced by the Dirichlet form of a general symmetric Markov process. A number of examples are provided.