On the Poles of Igusa's Local Zeta Function for Algebraic Sets

Let K be a p-adic field, let Z (s, f), sC, with Re(s) > 0,be the Igusa local zeta function associated to f(x) = (f₁(x),..., f_l(x)) [K (x₁, ..., x_n)]^l, and let be a Schwartz–Bruhatfunction. The aim of this paper is to describe explicitly thepoles of the meromorphic continuation of Z (s, f). Using resolutionof singularities it is possible to express Z (s, f) as a finitesum of p-adic monomial integrals. These monomial integrals arecomputed explicitly by using techniques of toroidal geometry.In this way, an explicit list of the candidates for poles ofZ (s, f) is obtained. 2000 Mathematics Subject Classification11S40, 14M25, 11D79.     

The Denef-Loeser Zeta Function is not a Topological Invariant

An example is given which shows that the Denef–Loeserzeta function (usually called the topological zeta function)associated to a germ of a complex hypersurface singularity isnot a topological invariant of the singularity. The idea isthe following. Consider two germs of plane curves singularitieswith the same integral Seifert form but with different topologicaltype and which have different topological zeta functions. Makea double suspension of these singularities (consider them ina 4-dimensional complex space). A theorem of M. Kervaire andJ. Levine states that the topological type of these new hypersurfacesingularities is characterized by their integral Seifert form.Moreover the Seifert form of a suspension is equal (up to sign)to the original Seifert form. Hence these new singularitieshave the same topological type. By means of a double suspensionformula the Denef–Loeser zeta functions are computed forthe two 3-dimensional singularities and it is verified thatthey are not equal.     

Poles of Zeta Functions on Normal Surfaces

Let (S, 0) be a normal surface germ and Let f a non-constantregular function on Let (S, 0) with Let f(0) = 0. Using anyadditive invariant on complex algebraic varieties one can associatea zeta function to these data, where the topological and motiviczeta functions are the roughest and the finest zeta functions,respectively. In this paper we are interested in a geometricdetermination of the poles of these functions. The second authorhas already provided such a determination for the topologicalzeta function in the case of non-singular surfaces. Here wegive a complete answer for all normal surfaces, at least onthe motivic level. The topological zeta function however seemsto be too rough for this purpose, although for negative poles,which are the only ones in the non-singular case, we are ableto prove exactly the same result as for non-singular surfaces. We also give and verify a (natural) definition for when a rationalnumber is a pole of the motivic zeta function. 2000 MathematicsSubject Classification 14B05, 14E15, 14J17 (primary), 32S50(secondary).     

关于Nielson Zeta函数的有理性

Ａ．Ｌ．Ｆｅｌ＇ｓｈｔｙｎ和Ｖ．Ｂ．Ｐｉｌｙｕｇｉｎａ定义了Ｎｉｅｌｓｏｎｚｅｔａ函数并证明关于其有理性的一些结果，本文改进了他们的一些结果。     

On the Probabilistic Zeta Function for Finite Groups

LetGbe a finite group, and define the function[formula]where μ is the Möbius function on the subgroup lattice ofG. The functionP(G, s) is the multiplicative inverse of a zeta function forG, as described by Mann and Boston. Boston conjectured thatP′(G, 1) = 0 ifGis a nonabelian simple. We will prove a generalization of this conjecture, showing thatP′(G, 1) = 0 unlessG/O_p(G) is cyclic for some primep.     

On Finite Maximal Antichains in the Homomorphism Order

We show that for structures with at most two relations all finite maximal antichains in the homomorphism order correspond to finite homomorphism dualities. We also show that most finite maximal antichains in this order split.     

Maximal Classes of Topological Spaces and Domains Determined by Function Spaces

In this paper the question of what classes A of T ₀-spaces should be paired with classes of domains in order that all function spaces [AB] for AA and B are -compact domains is considered. It is shown that core compact spaces are paired with bounded complete domains and a class of topological spaces called RW-spaces (with finitely many components) is paired with the class of -compact pointed L-domains (L-domains).     

黎曼ζ-函数之一:KS-变换——献给杨乐先生八十华诞

我们定义了KS-变换和自然数乘法结构相关的Fourier变换,建立了实数乘法半群[1,∞)={x:x∈R,x≥1}和复半平面Ω={s=σ+it:σ,t∈R,σ≥1/2}之间的由KS-变换诱导的对偶关系,证明了KS-变换是希尔伯特空间L~2([1,∞))和哈代空间H~2(Ω)之间的等距算子,而且该算子保持了相关的函数空间之间由实数的乘法卷积和复数点点相乘诱导出的代数结构的同构.作为应用,我们给出了黎曼假设成立的有关算子指标的等价命题,从而算子理论为研究黎曼ζ-函数和自然数的乘法结构提供了新思路.     

On the Ihara Zeta Function of Cones Over Regular Graphs

In this paper we derive a formula of the Ihara zeta function of a cone over a regular graph that involves the spectrum of the adjacency matrix of the cone. We show that the Ihara zeta function and the spectrum of the adjacency matrix of the cone determine each other and we characterize those cones that satisfy the graph theory Riemann hypothesis.     

On Some Integrals Involving the Hurwitz Zeta Function: Part 2

We establish a series of indefinite integral formulae involving the Hurwitz zeta function and other elementary and special functions related to it, such as the Bernoulli polynomials, ln sin(q), ln (q) and the polygamma functions. Many of the results are most conveniently formulated in terms of a family of functions A _k(q) := k(1 – k, q), k , and a family of polygamma functions of negative order, whose properties we study in some detail.     

On Some Integrals Involving the Hurwitz Zeta Function: Part 1

We establish a series of integral formulae involving the Hurwitz zeta function. Applications are given to integrals of Bernoulli polynomials, ln (q) and ln sin(q).     

Sums Involving the Hurwitz Zeta Function

We shall prove a general closed formula for integrals considered by Ramanujan, from which we derive our former results on sums involving Hurwitz zeta-function in terms not only of the derivatives of the Hurwitz zeta-function, but also of the multiple gamma function, thus covering all possible formulas in this direction. The transition from the derivatives of the Hurwitz zeta-function to the multiple gamma function and vice versa is proved to be effected essentially by the orthogonality relation of Stirling numbers.     

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.     

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.     

Monotonicity Properties of the Riemann Zeta Function

Let $$F_{a}(s) = \left(1 - 1\frac{1}{\zeta(s)}\right)^{1/(s-a)}(a \leq 1; s > 1),$$ where ?? denotes the Riemann zeta function. We prove: F _a is strictly decreasing on (1, ??) if and only if a ?? 0, whereas F _a is strictly increasing on (1, ??) if and only if a =?1. In particular, this settles a conjecture of Batir, who claimed that F ₀ is strictly monotonic for s >?1. Moreover, we apply the monotonicity theorem to obtain some inequalities involving F _a .     

A Multicomplex Riemann Zeta Function

After reviewing properties of analytic functions on the multicomplex number space ${\mathbb{C}_{k}}$ (a commutative generalization of the bicomplex numbers ${\mathbb{C}_{2}}$ ), a multicomplex Riemann zeta function is defined through analytic continuation. Properties of this function are explored, and we are able to state a multicomplex equivalence to the Riemann hypothesis.     

On the Shintani Zeta Function for the Space of Pairs of Binary Hermitian Forms

In this paper we determine the principal part of the adjusted zeta function for the space of pairs of binary Hermitian forms.