关于一个数论函数的导数及应用

Kanemitsu教授给出了欧拉求和函数的推广公式Lu(x,a)=0n相似文献   

A generalization of the riemann zeta-function

A generalization of the Riemann zeta-function which has the form

The Universality of Zeta-Functions

The first part of the paper contains a survey on the universality of zeta-functions. Zeta-functions with Euler's product as well as zeta-functions without Euler's product are discussed. Also, the joint universality theorems are considered. In the second part of the paper the universality of zeta-functions of finite Abelian groups of rank 3 is proved.     

关于Smarandache函数与Riemann zeta-函数

对任意正整数n,著名的Smarandache函数S(n)定义为最小的正整数m使得n|m!.即S(n)=min{m∶m ∈N,n|m!).本文的主要目的是利用初等方法研究一类包含S(n)的Dirichlet级数与Riemann zeta-函数之间的关系,并得到了一个有趣的恒等式.     

Chamber behavior of double Hurwitz numbers in genus 0

We study double Hurwitz numbers in genus zero counting the number of covers CP¹→CP¹ with two branching points with a given branching behavior. By the recent result due to Goulden, Jackson and Vakil, these numbers are piecewise polynomials in the multiplicities of the preimages of the branching points. We describe the partition of the parameter space into polynomiality domains, called chambers, and provide an expression for the difference of two such polynomials for two neighboring chambers. Besides, we provide an explicit formula for the polynomial in a certain chamber called totally negative, which enables us to calculate double Hurwitz numbers in any given chamber as the polynomial for the totally negative chamber plus the sum of the differences between the neighboring polynomials along a path connecting the totally negative chamber with the given one.     

Equations of Hurwitz schemes in the infinite Grassmannian

The main result proved in the paper is the computation of the explicit equations defining the Hurwitz schemes of coverings with punctures as subschemes of the Sato infinite Grassmannian. As an application, we characterize the existence of certain linear series on a smooth curve in terms of soliton equations. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

An efficient algorithm for accelerating the convergence of oscillatory series,useful for computing the polylogarithm and Hurwitz zeta functions

This paper sketches a technique for improving the rate of convergence of a general oscillatory sequence, and then applies this series acceleration algorithm to the polylogarithm and the Hurwitz zeta function. As such, it may be taken as an extension of the techniques given by Borwein’s “An efficient algorithm for computing the Riemann zeta function” by Borwein for computing the Riemann zeta function, to more general series. The algorithm provides a rapid means of evaluating Li _s (z) for general values of complex s and a kidney-shaped region of complex z values given by ∣z ²/(z–1)∣<4. By using the duplication formula and the inversion formula, the range of convergence for the polylogarithm may be extended to the entire complex z-plane, and so the algorithms described here allow for the evaluation of the polylogarithm for all complex s and z values. Alternatively, the Hurwitz zeta can be very rapidly evaluated by means of an Euler–Maclaurin series. The polylogarithm and the Hurwitz zeta are related, in that two evaluations of the one can be used to obtain a value of the other; thus, either algorithm can be used to evaluate either function. The Euler–Maclaurin series is a clear performance winner for the Hurwitz zeta, while the Borwein algorithm is superior for evaluating the polylogarithm in the kidney-shaped region. Both algorithms are superior to the simple Taylor’s series or direct summation. The primary, concrete result of this paper is an algorithm allows the exploration of the Hurwitz zeta in the critical strip, where fast algorithms are otherwise unavailable. A discussion of the monodromy group of the polylogarithm is included.      

On Balazard,Saias, and Yor’s equivalence to the Riemann Hypothesis

Balazard, Saias, and Yor proved that the Riemann Hypothesis is equivalent to a certain weighted integral of the logarithm of the Riemann zeta-function along the critical line equaling zero. Assuming the Riemann Hypothesis, we investigate the rate at which a truncated version of this integral tends to zero, answering a question of Borwein, Bradley, and Crandall and disproving a conjecture of the same authors. A simple modification of our techniques gives a new proof of a classical Omega theorem for the function S(t)$S\left(t\right)$ in the theory of the Riemann zeta-function.     

Electromechanical control of vibration on a plate submitted to a non-ideal excitation

This paper deals with the enhancement of electromechanical control of vibration on a thin plate submitted to non-ideal excitation. Modelling of the systems displays the non-ideal source used as external excitation above the structure on a particular surface and control force acting at specific points under the structure. The electromechanical device is composed by a RL circuit with a saturated inductance and stings connected to the plate is used as connection between the structure and the controller. Routh–Hurwitz criteria are used to obtain the stability condition of the controlled system and some dynamics exploration leads us to the condition for which the amplitude of vibration is reduced in the mechanical structure.     

The joint value-distribution of the Riemann zeta function and Hurwitz zeta functions

In this paper, we investigate the joint value-distribution for the Riemann zeta function and Hurwitz zeta function attached with a transcendental real parameter. Especially, we establish the joint universality theorem for these two zeta functions. Published in Lietuvos Matematikos Rinkinys, Vol. 47, No. 1, pp. 39–57, January–March, 2007.