Consistent first-return Riemann sums for Lebesgue integrals

U. B. Darji and M. J. Evans [1] showed previously that it is possible to obtain the integral of a Lebesgue integrable function on the interval [0,1] via a Riemann type process, where one chooses the selected point in each partition interval using a first-return algorithm based on a sequence {x _n} which is dense in [0,1]. Here we show that if the same is true for every rearrangement of {x _n}, then the function must be equal almost everywhere to a Riemann integrable function. This revised version was published online in June 2006 with corrections to the Cover Date.     

关于Genocchi数和Riemann Zeta-函数的一些恒等式

利用计算技巧给出了由Ｇｅｎｏｃｃｉ数和ＲｉｅｍａｎｎＺｅｔａ－函数组成的和式的递归关系，得到了一些关于Ｇｅｎｏｃｃｈｉ数和ＲｉｅｍａｎｎＺｅｔａ－函数的恒等式     

Riemann Zeta函数的七阶和式

通过构造一个Riemann Zeta函数ζ(k)的部分和ζ_n(k)的幂级数函数,利用牛顿二项式展开及柯西乘积公式可以计算出一些重要的和式.再将该幂级数函数由一元推广到二元甚至多元,由此得到Riemann Zeta函数的高次方和式之间的关系.并利用对数函数与第一类Stirling数之间的关系式及ζ(k)函数满足的相关等式,可得出Riemann Zeta函数的18个七阶和式,以及其它一些高次方的和式.     

关于Genocchi数和Riemann Zeta-函数的一些恒等式

利用计算技巧给出了由Ｇｅｎｏｃｃｉ数和Ｒｉｃｍａｎｎ Ｚｅｔａ－函数组成的和式的递归关系，得到了一些关于Ｇｅｎｏｃｃｈｉ Ｚｅｔａ－函数的恒等式。     

The Igusa Zeta Function Associated with a Composite Power Function on the Space of Rectangular Matrices

On the space of real rectangular n × m matrices, we introduce a composite power function and study the zeta integral associated with it. We describe the properties of the Igusa zeta function on the basis of the properties of a generalized composite power function and establish a functional relation for the zeta integral. As a result, the Fourier transform of a generalized composite power function is found in explicit form.     

Omega Theorems for Zeta Sums

We obtain lower bounds for the moduli of trigonometric sums in the theory of Riemann zeta functions.     

Continued-fraction expansions for the Riemann zeta function and polylogarithms

It appears that the only known representations for the Riemann zeta function in terms of continued fractions are those for and 3. Here we give a rapidly converging continued-fraction expansion of for any integer . This is a special case of a more general expansion which we have derived for the polylogarithms of order , , by using the classical Stieltjes technique. Our result is a generalisation of the Lambert-Lagrange continued fraction, since for we arrive at their well-known expansion for . Computation demonstrates rapid convergence. For example, the 11th approximants for all , , give values with an error of less than 10.

     

Lower Bounds for the Riemann Zeta Function on the Critical Line

We establish a relation between the lower bound for the maximum of the modulus of $\zeta (1/2 + iT + s)$ in the disk $|s| \leqslant H$ and the lower bound for the maximum of the modulus of $\zeta (1/2 + iT + it)$ on the closed interval $|t| \leqslant H$ for $0 < H(T) \leqslant {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0em} 2}$ . We prove a theorem on the lower bound for the maximum of the modulus of $0 < H(T) \leqslant {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0em} 2}$ on the closed interval $|t| \leqslant H$ for $40 \leqslant H(T) \leqslant \log \log T$ .     

Zeta Regularizations

We survey our work on various generalizations of the usual zeta regularized product and their applications. For construction of functions having a certain invariance and a given set of zeros, so-called the zeta regularization method is quite useful. We exhibit several examples and discuss some questions.     

Exponential Sums and the Riemann Zeta Function V

A Van der Corput exponential sum is S = exp (2 i f(m)) wherem has size M, the function f(x) has size T and = (log M) / log T < 1. There are different bounds for S in differentranges for . In the middle range where is near 1/over 2, . This bounds the exponent of growthof the Riemann zeta function on its critical line Re s = 1/over2. Van der Corput used an iteration which changed at each step.The Bombieri–Iwaniec method, whilst still based on meansquares, introduces number-theoretic ideas and problems. TheSecond Spacing Problem is to count the number of resonancesbetween short intervals of the sum, when two arcs of the graphof y = f'(x) coincide approximately after an automorphism ofthe integer lattice. In the previous paper in this series [Proc.London Math. Soc. (3) 66 (1993) 1–40] and the monographArea, lattice points, and exponential sums we saw that coincidenceimplies that there is an integer point close to some ‘resonancecurve’, one of a family of curves in some dual space,now calculated accurately in the paper ‘Resonance curvesin the Bombieri–Iwaniec method’, which is to appearin Funct. Approx. Comment. Math. We turn the whole Bombieri–Iwaniec method into an axiomatisedstep: an upper bound for the number of integer points closeto a plane curve gives a bound in the Second Spacing Problem,and a small improvement in the bound for S. Ends and cusps ofresonance curves are treated separately. Bounds for sums oftype S lead to bounds for integer points close to curves, andanother branching iteration. Luckily Swinnerton-Dyer's methodis stronger. We improve from 0.156140... in the previous paperand monograph to 0.156098.... In fact (32/205 + , 269/410 +) is an exponent pair for every > 0. 2000 Mathematics SubjectClassification 11L07 (primary), 11M06, 11P21, 11J54 (secondary).     

Some Unusual Identities for Special Values of the Riemann Zeta Function

In this paper, we use elementary methods to derive some new identities for special values of the Riemann zeta function.     

Hermite Expansion of the Riemann Zeta Function

Let $ζ(s)$ be the Riemann zeta function, $s=\sigma+it$. For $0 < \sigma < 1$, we expand $ζ(s)$ as the following series convergent in the space of slowly increasing distributions with variable $t$ : $$ζ(\sigma+it)=\sum\limits^∞_{n=0}a_n(\sigma)ψ_n(t),$$ where $$ψ_n(t)=(2^nn!\sqrt{\pi})^{-1 ⁄ 2}e^{\frac{-t^2}{2}}H_n(t),$$ $H_n(t)$ is the Hermite polynomial, and $$a_n(σ)=2\pi(-1)^{n+1}ψ_n(i(1-σ))+(-i)^n\sqrt{2\pi}\sum\limits^∞_{m=1}\frac{1}{m^σ}ψ_n(1nm).$$ This paper is concerned with the convergence of the above series for $σ > 0.$ In the deduction, it is crucial to regard the zeta function as Fourier transfomations of Schwartz' distributions.     

Riemann Zeta函数的六阶和

利用概率论与组合数学的方法,研究了与Riemann-zeta函数ξ(k)的部分和ξ_n(k)有关的一些级数,计算出了一些重要的和式．特别的,Euler的著名结果5ξ(4)= 2ξ~2(2)能够从四阶和式直接推出．因此,通过计算全部的11个六阶和式,研究它们之间的非平凡关系,就有可能得到ξ(3)的数值．     

An Approximate Functional Equation for the Lerch Zeta Function

In this paper, we establish an approximate functional equation for the Lerch zeta function, which is a generalization of the Riemann zeta function and the Hurwitz zeta function.     

On Functional Relations for the Alternating Analogues of Tornheim's Double Zeta Function

In this paper, new proofs of two functional relations for the alternating analogues of Tornheim's double zeta function are given. Using the functional relations, the author gives new proofs of some evaluation formulas found by Tsumura for these alternating series.     

Transmission Eigenvalues and the Riemann Zeta Function in Scattering Theory for Automorphic Forms on Fuchsian Groups of Type I

We introduce the concept of transmission eigenvalues in scattering theory for automorphic forms on fundamental domains generated by discrete groups acting on the hyperbolic upper half complex plane. In particular, we consider Fuchsian groups of type Ⅰ. Transmission eigenvalues are related to those eigen-parameters for which one can send an incident wave that produces no scattering. The notion of transmission eigenvalues, or non-scattering energies, is well studied in the Euclidean geometry, where in some cases these eigenvalues appear as zeros of the scattering matrix. As opposed to scattering poles, in hyperbolic geometry such a connection between zeros of the scattering matrix and non-scattering energies is not studied, and the goal of this paper is to do just this for particular arithmetic groups. For such groups, using existing deep results from analytic number theory, we reveal that the zeros of the scattering matrix, consequently non-scattering energies, are directly expressed in terms of the zeros of the Riemann zeta function. Weyl's asymptotic laws are provided for the eigenvalues in those cases along with estimates on their location in the complex plane.     

A Lower Bound in an Approximation Problem Involving the Zeros of the Riemann Zeta Function

We slightly improve the lower bound of Báez-Duarte, Balazard, Landreau and Saias in the Nyman-Beurling formulation of the Riemann Hypothesis as an approximation problem. We construct Hilbert space vectors which could prove useful in the context of the so-called “Hilbert-Pólya idea”.     

黎曼积分的完备化

综述了黎曼可积函数的基本特征,并指出黎曼可积函数列的极限运算在积分意义下是不封闭的.在构造了完备化空间之后,证明了该空间就是勒贝格可积函数空间,从而说明了黎曼积分的完备化形式是勒贝格积分.     

Remark on a double-inequality for the Riemann zeta function

Let ζ be the Riemann zeta function and δ(x)=1/(2^x-1). For all x>0 we have

(1-δ(x))ζ(x)+αδ(x)<ζ(x+1)<(1-δ(x))ζ(x)+βδ(x),