Sabir M. Gusein-Zade, Igancio Luengo, Alejandro Melle-Hernández
For a germ of a meromorphic function f=P/Q, we offer notions of the monodromy operators at zero and at infinity. If the holomorphic functions P and Q are non-degenerated with respect to their Newton diagrams, we give an analogue of the formula of Varchenko for the zeta-functions of these monodromy operators.
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The zeta function of zeros and poles of a meromorphic function, the criterion for the absence of zeros, and its application to sums of Cauchy kernels
该研究通过联系亚纯函数零点与极点的 zeta 函数，提出零点不存在判据，并给出 Cauchy 核之和零点存在的充分条件。
On the algebraicity of germs of meromorphic functions
该研究证明复二维空间原点处的解析亚纯函数芽可通过双全纯芽右复合等价于代数亚纯函数芽，且实解析情形下也有类似结果，推进了亚纯函数芽代数性相关理论。
We prove that a meromorphic mapping, which sends a peace of a real analytic strictly pseudoconvex hypersurface in $\cc^2$ to a compact subset of $\cc^N$ which doesn't contain germs of non-constant complex curves is continuous from the concave side of the hypersurface. This implies the analytic continuability along CR-paths of germs of holomorphic mappings from real analytic hypersurfaces with non-vanishing Levi form to the locally spherical ones in all dimensions.
Meromorphic continuation and non-polar singularities of local zeta functions in some smooth cases
本文研究了特定非实解析光滑函数的局部 zeta 函数，证明了其在某些奇点处具有非极点奇点，并验证了特定光滑函数类下亚纯延拓下界估计的最优性。
We investigate the existence of the meromorphic extension of the spectral zeta function of the Laplacian on self-similar fractals using the classical results of Kigami and Lapidus (based on the renewal theory) and new results of Hambly and Kajino based on the heat kernel estimates and other probabilistic techniques. We also formulate conjectures which hold true in the examples that have been analyzed in the existing literature.
Meromorphic germs in several variables with linear poles naturally arise in mathematics in various disguises. We investigate their rich structures under the prism of locality, including locality subalgebras, locality transformation groups and locality characters. The key technical tool is the dependence subspace for a meromorphic germ with which we define a locality orthogonal relation between two meromorphic germs. We describe the structure of locality subalgebras generated by classes of meromorphic germs with certain types of poles. We also define and determine their group of locality transf…
A topological splitting of the space of meromorphic germs in several variables and continuous evaluators
本文在Silva空间框架下利用内积，将多变量零点处亚纯芽空间拓扑分解为全纯芽与极芽之和，并由此构造零点处的连续赋值子，推广并细化了已有分解结果。
It is known that local zeta functions associated with real analytic functions can be analytically continued as meromorphic functions to the whole complex plane. But, in the case of general ($C^{\infty}$) smooth functions, the meromorphic extension problem is not obvious. Indeed, it has been recently shown that there exist specific smooth functions whose local zeta functions have singularities different from poles. In order to understand the situation of the meromorphic extension in the smooth case, we investigate a simple but essentially important case, in which the respective function is expr…
We use convex polyhedral cones to study a large class of multivariate meromorphic germs, namely those with linear poles, which naturally arise in various contexts in mathematics and physics. We express such a germ as a sum of a holomorphic germ and a linear combination of special non-holomorphic germs called polar germs. In analyzing the supporting cones -- cones that reflect the pole structure of the polar germs -- we obtain a geometric criterion for the non-holomorphicity of linear combinations of polar germs. This yields the uniqueness of the above sum when required to be supported on a sui…
Two semigroup rings associated to a finite set of germs of meromorphic functions
本文研究由有限个亚纯函数芽生成的半群环，给出其一般性质，并详细分析阶小于1的亚纯函数域及有限阶、有限零极点情形的结构。
Quasiperiodic sets at infinity and meromorphic extensions of their fractal zeta functions
本文研究无穷远处相对分形鼓的复维数，建立管状与距离zeta函数的函数方程，给出亚纯延拓、Minkowski可测性条件及上容度界，并证明无穷远处准周期集合复维数的代数或超越准周期结构。
We consider the problem of extending germs of plane holomorphic foliations to foliations of compact surfaces. We show that the germs that become regular after a single blow up and admit meromorphic first integrals can be extended, after local changes of coordinates, to foliations of compact surfaces. We also show that the simplest elements in this class can be defined by polynomial equations. On the other hand we prove that, in the absence of meromorphic first integrals there are uncountably many elements without polynomial representations.
In \cite{BeEr}, Bergweiler and Eremenko computed the number of critical points of the Green's function on a torus by investigating the dynamics of a certain family of antiholomorphic meromorphic functions on tori. They also observed that hyperbolic maps are dense in this family of meromorphic functions in a rather trivial way. In this paper, we study the parameter space of this family of meromorphic functions, which can be written as antiholomorphic perturbations of Weierstrass Zeta functions. On the one hand, we give a complete topological description of the hyperbolic components and their bo…
In this paper we treat the classical Riemann zeta function as a function of three variables: one is the usual complex $\adyn$-dimensional, customly denoted as $s$, another two are complex infinite dimensional, we denote it as $\b = \{b_n\}_{n=1}^{\infty}$ and $\z =\{z_n\}_{n=1}^{\infty}$. When $\b = \{1\}_{n=1}^{\infty}$ and $\z = \{\frac{1}{n}\}_{n=1}^{\infty}$ one gets the usual Riemann zeta function. Our goal in this paper is to study the meromorphic continuation of $ζ(\b , \z ,s)$ as a function of the triple $(\a , \z , s)$. Minor corrections, to appear in the Journal of Mathematical Analy…
For an arbitrary tuple of $m+1$ germs of analytic functions at a fixed point, we introduce the so-called polynomial Hermite-Padé $m$-system (of order $n$, $n\in\mathbb N$), which consists of $m$ tuples of polynomials; these tuples, which are indexed by a natural number $k\in[1,\dots,m]$, are called the $k$th polynomials of the Hermite-Padé $m$-system. We study the weak asymptotics of the polynomials of the Hermite-Padé $m$-system constructed at the point $\infty$ from the tuple of germs $[1, f_{1,\infty},\dotsc$, $f_{m,\infty}]$ of the functions $1, f_1,\dots,f_m$ that are meromorphic on some …
It is proved that the Riemann zeta function does not satisfy any nontrivial algebraic difference equation whose coefficients are meromorphic functions $ϕ$ with Nevanlinna characteristic satisfying $T(r, ϕ)=o(r)$ as $r\to \infty$
An exp-algebraic curve consists of a compact Riemann surface $S$ together with $n$ equivalence classes of germs of meromorphic functions modulo germs of holomorphic functions, $\HH = \{ [h_1], \cdots, [h_n] \}$, with poles of orders $d_1, \cdots, d_n \geq 1$ at points $p_1, \cdots, p_n$. This data determines a space of functions $\OO_{\HH}$ (respectively, a space of $1$-forms $Ω^0_{\HH}$) holomorphic on the punctured surface $S' = S - \{p_1, \cdots, p_n\}$ with exponential singularities at the points $p_1, \cdots, p_n$ of types $[h_1], \cdots, [h_n]$, i.e., near $p_i$ any $f \in \OO_{\HH}$ is …
We show that it is possible to approximate the zeta-function of a curve over a finite field by meromorphic functions which satisfy the same functional equation and moreover satisfy (respectively do not satisfy) the analogue of the Riemann hypothesis. In the other direction, it is possible to approximate holomorphic functions by simple manipulations of such a zeta-function. We also consider the value distribution of zeta-functions of function fields over finite fields from the viewpoint of Nevanlinna theory.
Logarithmic models and meromorphic functions in dimension two
本文研究二维空间中对数模型的构造及其在亚纯函数和向量场中的应用，通过分离子和Camacho-Sad指数构建具有特定奇点结构的亚纯函数。
The random Weierstrass zeta function I. Existence, uniqueness, fluctuations
本文提出一种在给定平稳点过程处具有单位留数简单极点的随机亚纯函数构造方法，刻画了使其减去均值后平稳的有限二阶矩条件，并将其视为 Weierstrass zeta 函数的随机类比或随机点电荷电场。
Laplace measure transitions and ghosts for meromorphic functions
研究双侧拉普拉斯变换在垂直条带上的测度转换问题，探讨确定性测度的关系，推导有限和无限极点情况下的转换公式，并应用于ζ函数的解析延拓及其它函数的周期性分析。
It is known that local zeta functions associated with real analytic functions can be analytically continued as meromorphic functions to the whole complex plane. In this paper, the case of specific (non-real analytic) smooth functions is precisely investigated. Indeed, asymptotic limits of the respective local zeta functions at some singularities in one direction are explicitly computed. Surprisingly, it follows from these behaviors that these local zeta functions have singularities different from poles.
In this paper we establish a close connection between three notions at- tached to a modular subgroup. Namely the set of weight two meromorphic modular forms, the set of equivariant functions on the upper half-plane commuting with the action of the modular subgroup and the set of elliptic zeta functions generalizing the Weierstrass zeta functions. In particular, we show that the equivariant functions can be parameterized by modular objects as well as by elliptic objects.
A family of Zeta functions built as Dirichlet series over the Riemann zeros are shown to have meromorphic extensions in the whole complex plane, for which numerous analytical features (the polar structure, plus countably many special values) are explicitly displayed.
In this paper, we consider linear $q$-difference systems with coefficients that are germs of meromorphic functions, with Newton polygon that has two slopes. Then, we explain how to compute similar meromorphic gauge transformations than those appearing in the work of Bugeaud, using a $q$-analogue of the Borel-Laplace summation.
We prove that the zeta-function $ζ_Δ$ of the Laplacian $Δ$ on a self-similar fractals with spectral decimation admits a meromorphic continuation to the whole complex plane. We characterise the poles, compute their residues, and give expressions for some special values of the zeta-function. Furthermore, we discuss the presence of oscillations in the eigenvalue counting function.
Mittag-Leffler type theorems for Helson zeta-functions
本文证明了 Helson zeta-函数可亚纯延拓至复平面并具有预定零极点，改进了先前仅在特定带形区域内的结果，并揭示了其亚纯性与解析性的最大可能区域。
The theory of 'zeta functions of fractal strings' has been initiated by the first author in the early 1990s, and developed jointly with his collaborators during almost two decades of intensive research in numerous articles and several monographs. In 2009, the same author introduced a new class of zeta functions, called `distance zeta functions', which since then, has enabled us to extend the existing theory of zeta functions of fractal strings and sprays to arbitrary bounded (fractal) sets in Euclidean spaces of any dimension. A natural and closely related tool for the study of distance zeta f…
The aim of this work is to study the analytic continuation of the doubly-periodic Barnes zeta function. By using a suitable complex integral representation as a starting point we find the meromorphic extension of the doubly periodic Barnes zeta function to the entire complex plane in terms of a real integral containing the Hurwitz zeta function and the first Jacobi theta function. These allow us to explicitly give expressions for the derivative at all non-positive integer points.
The modified zeta functions $\sum_{n \in K} n^{-s}$, where $K \subset \N$, converge absolutely for $\Re s > 1/2$. These generalise the Riemann zeta function which is known to have a meromorphic continuation to all of $\C$ with a single pole at $s=1$. Our main result is a characterisation of the modified zeta functions that have pole-like behaviour at this point. This behaviour is defined by considering the modified zeta functions as kernels of certain integral operators on the spaces $L^2(I)$ for symmetric and bounded intervals $I \subset \R$. We also consider the special case when the set $K …
Bernstein-Sato polynomial and related invariants for meromorphic functions
本文建立亚纯函数的 Bernstein-Sato 多项式理论，用于分析阿基米德局部 zeta 函数的解析延拓，并提出乘子理想的解析与代数理论，揭示其跳跃数与多项式根的关系。
It is known that Shintani zeta functions, which generalise multiple zeta functions, extend to meromorphic functions with poles on affine hyperplanes. We refine this result in showing that the poles lie on hyperplanes parallel to the facets of certain convex polyhedra associated to the defining matrix for the Shintani zeta function. Explicitly, the latter are the Newton polytopes of the polynomials induced by the columns of the underlying matrix. We then prove that the coefficients of the equation which describes the hyperplanes in the canonical basis are either zero or one, similar to the pole…
We consider the series $\sum_{n=1}^{\infty} z^{n} (a_{n} + x)^{-s}$ where $a_{n}$ satisfies a linear recurrence of arbitrary degree with integer coefficients. Under appropriate conditions, we prove that it can be continued to a meromorphic function on the complex $s$-plane. Thus we may associate a Lerch-type zeta function $φ(z,s,x)$ to a general recurrence. This subsumes all previous results which dealt only with the ordinary zeta and Hurwitz cases and degrees $2$ and $3$. Our method generalizes a formula of Ramanujan for the classical Hurwitz-Riemann zeta functions. We determine the poles and…
We study the theta function and the Hurwitz-type zeta function associated to the Lucas sequence $U=\{U_n(P,Q)\}_{n\geq 0}$ of the first kind determined by the real numbers $P,Q$ under certain natural assumptions on $P$ and $Q$. We deduce an asymptotic expansion of the theta function $θ_U(t)$ as $t\downarrow 0$ and use it to obtain a meromorphic continuation of the Hurwitz-type zeta function $ζ_{U}\left( s,z\right) =\sum\limits_{n=0}^{\infty }\left(z+U_{n}\right) ^{-s}$ to the whole complex $s-$plane. Moreover, we identify the residues of $ζ_{U}\left( s,z\right)$ at all poles in the half-plane …
Fractal zeta functions of orbits of parabolic diffeomorphisms
本文证明抛物型微分同胚芽轨道的分形zeta函数可亚纯延拓至全复平面，并通过其极点与主部刻画轨道的分形足迹，进而提取抛物芽的形式类等内在信息，同时揭示这类轨道具有非平凡盒维数和高阶振荡管函数但无复维数的独特性质。
We study the incomplete Mellin transformation of the fractional part and the related log-sine function when composed by an affine complex map. We evaluate the corresponding integral in two different ways which yields equalities with series in hypergeometric functions on each side. These equalities capture basic analytic properties of the Hurwitz zeta function like meromorphic extension and the functional equation. We give rates of convergence for the involved series. As a special case we find that the exponential rate of convergence for the deformation of the log-sine to the fractional part in…
This paper studies zeta functions of the form $\sum_{n=1}^{\infty} χ(n) n^{-s}$, with $χ$ a completely multiplicative function taking only unimodular values. We denote by $σ(χ)$ the infimum of those $α$ such that the Dirichlet series $\sum_{n=1}^{\infty} χ(n) n^{-s}$ can be continued meromorphically to the half-plane $\operatorname{Re} s>α$, and denote by $ζ_χ(s)$ the corresponding meromorphic function in $\operatorname{Re} s>σ(χ)$. We construct $ζ_χ(s)$ that have $σ(χ)\le 1/2$ and are universal for zero-free analytic functions on the half-critical strip $1/2<\operatorname{Re} s <1$, with zero…
On Milnor and Tjurina numbers of Pairs of Holomorphic Functions Germs
该研究受全纯函数芽对的米尔诺数分岔公式启发，通过叶状理论引入蒂尤林数，证明其为解析不变量并推导公式及性质，还用此不变量刻画半驯亚纯函数芽，同时研究了全纯函数芽对的米尔诺数相关性质。
Resolution of singularities for $C^{\infty}$ functions and meromorphy of local zeta functions
本文通过构造普通爆破算法将二元$C^{\infty}$函数零点簇化为几乎法向交叉形式，并借助van der Corput型引理，用零点簇的重数刻画了相应局部zeta函数的亚纯延拓区域。
We construct a large family of Fourier interpolation bases for functions analytic in a strip symmetric about the real line. Interesting examples involve the nontrivial zeros of the Riemann zeta function and other $L$-functions. We establish a duality principle for Fourier interpolation bases in terms of certain kernels of general Dirichlet series with variable coefficients. Such kernels admit meromorphic continuation, with poles at a sequence dual to the sequence of frequencies of the Dirichlet series, and they satisfy a functional equation. Our construction of concrete bases relies on a stren…
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