On Uniqueness of Dirichlet Series

2020 ◽  
Author(s):  
Bao Qin Li
Keyword(s):  
Analytic Continuation ◽  
Dirichlet Series ◽  
Finite Order ◽  
Half Plane ◽  
Complex Plane ◽  
Meromorphic Functions ◽  
Uniqueness Problem ◽  
Uniqueness Theorems

Abstract We give a characterization of the ratio of two Dirichelt series convergent in a right half-plane under an analytic condition, which is applicable to a uniqueness problem for Dirichlet series admitting analytic continuation in the complex plane as meromorphic functions of finite order; uniqueness theorems are given in terms of a-points of the functions.

scholarly journals Logarithmic derivative estimates of meromorphic functions of finite order in the half-plane

2020 ◽  
Vol 54 (2) ◽  
pp. 172-187
Author(s):  
I.E. Chyzhykov ◽  
A.Z. Mokhon'ko
Keyword(s):  
Meromorphic Function ◽  
Finite Order ◽  
Half Plane ◽  
Meromorphic Functions ◽  
Logarithmic Derivative ◽  
Sharp Estimates ◽  
Exceptional Sets ◽  
Upper Half Plane ◽  
Logarithmic Derivatives

We established new sharp estimates outside exceptional sets for of the logarithmic derivatives $\frac{d^ {k} \log f(z)}{dz^k}$ and its generalization $\frac{f^{(k)}(z)}{f^{(j)}(z)}$, where $f$ is a meromorphic function $f$ in the upper half-plane, $k>j\ge0$ are integers. These estimates improve known estimates due to the second author in the class of meromorphic functions of finite order.Examples show that size of exceptional sets are best possible in some sense.

scholarly journals Reflection Identities of Harmonic Sums of Weight Four

Universe ◽  
2019 ◽  
Vol 5 (3) ◽  
pp. 77 ◽  
Author(s):  
Alexander Prygarin
Keyword(s):  
Analytic Continuation ◽  
Linear Combination ◽  
Complex Plane ◽  
Meromorphic Functions ◽  
Bfkl Equation ◽  
Color Singlet ◽  
Pole Structure

In attempt to find a proper space of function expressing the eigenvalue of the color-singlet BFKL equation in N = 4 SYM, we consider an analytic continuation of harmonic sums from positive even integer values of the argument to the complex plane. The resulting meromorphic functions have pole singularities at negative integers. We derive the reflection identities for harmonic sums at weight four decomposing a product of two harmonic sums with mixed pole structure into a linear combination of terms each having a pole at either negative or non-negative values of the argument. The pole decomposition demonstrates how the product of two simpler harmonic sums can build more complicated harmonic sums at higher weight. We list a minimal irreducible set of bilinear reflection identities at weight four, which represents the main result of the paper. We also discuss how other trilinear and quadlinear reflection identities can be constructed from our result with the use of well known quasi-shuffle relations for harmonic sums.

scholarly journals Delange's characterization of the sine function

1974 ◽  
Vol 15 (1) ◽  
pp. 66-68 ◽  
Author(s):  
Chin-Hung Ching ◽  
Charles K. Chui
Keyword(s):  
Entire Function ◽  
Analytic Continuation ◽  
Complex Plane ◽  
Real Line ◽  
Sine Function ◽  
The Real ◽  
Image Position

In [2], H. Delange gives the following characterization of the sine function.Theorem A. f(x)=sin x is the only infinitely differentiable real-valued function on the real line such that f'(O)= 1 andfor all real x and n = 0,1,2,….It is clear that, if f satisfies (1), then the analytic continuation of f is an entire function satisfyingfor all z in the complex plane. Hence f is of at most order one and type one. In this note, we prove the following theorem.

scholarly journals A note on the congruent distribution of the number of prime factors of natural numbers

2001 ◽  
Vol 163 ◽  
pp. 1-11 ◽  
Author(s):  
Tomio Kubota ◽  
Mariko Yoshida
Keyword(s):  
Analytic Continuation ◽  
Natural Number ◽  
Dirichlet Series ◽  
Half Plane ◽  
Arithmetic Function ◽  
Prime Numbers ◽  
Natural Numbers ◽  
Prime Factors

Let n = p1p2 … pr be a product of r prime numbers which are not necessarily different. We define then an arithmetic function µm(n) bywhere m is a natural number. We further define the function L(s, µm) by the Dirichlet seriesand will show that L(s, µm), (m ≥ 3), has an infinitely many valued analytic continuation into the half plane Re s > ½.

scholarly journals The uniqueness problem of meromorphic maps into the complex projective space

1975 ◽  
Vol 58 ◽  
pp. 1-23 ◽  
Author(s):  
Hirotaka Fujimoto
Keyword(s):  
Projective Space ◽  
Complex Plane ◽  
Complex Projective Space ◽  
Meromorphic Functions ◽  
Uniqueness Problem ◽  
Meromorphic Maps ◽  
Complex Projective

In 1921, G. Pólya showed that non-constant meromorphic functions ϕ and ψ of finite genera on the complex plane C are necessarily equal if there are distinct five values ai(1 ≦ i ≦ 5) such that ϕ(z) — ai and ψ(z) — ai have the same zeros of the same multiplicities for each i ([8]). Afterwards, R. Nevanlinna obtained the same conclusion for arbitrary ϕp and ψ satisfying ϕ— 1(ai) = ψ— 1(1 ≦ i ≦ 5) regardless of multiplicities. And, some other results relating to this were given by H. Cartan ([2], [3]), E. M. Schmid ([9]) and others. The purpose of this paper is to give some types of generalizations of these results to the case of meromorphic maps into the N-dimensional complex projective space PN(C).

scholarly journals Functional equation of a special Dirichlet series

1987 ◽  
Vol 10 (2) ◽  
pp. 395-403
Author(s):  
Ibrahim A. Abou-Tair
Keyword(s):  
Functional Equation ◽  
Holomorphic Function ◽  
Dirichlet Series ◽  
Half Plane ◽  
Complex Plane ◽  
Positive Integers

In this paper we study the special Dirichlet seriesL(s)=23∑n=1∞sin(2πn3)n−s,  s∈CThis series converges uniformly in the half-planeRe(s)>1and thus represents a holomorphic function there. We show that the functionLcan be extended to a holomorphic function in the whole complex-plane. The values of the functionLat the points0,±1,−2,±3,−4,±5,…are obtained. The values at the positive integers1,3,5,…are determined by means of a functional equation satisfied byL.

scholarly journals The analytic structure of the BFKL equation and reflection identities of harmonic sums at weight five

2019 ◽  
Vol 34 (11) ◽  
pp. 1950064 ◽  
Author(s):  
Mohammad Joubat ◽  
Alexander Prygarin
Keyword(s):  
Analytic Continuation ◽  
Complex Plane ◽  
Meromorphic Functions ◽  
Functional Form ◽  
Bfkl Equation ◽  
Analytic Structure ◽  
Color Singlet ◽  
Nonlinear Reflection ◽  
Pole Structure ◽  
Entire Complex

We analyze the structure of the eigenvalue of the color-singlet Balitsky–Fadin–Kuraev–Lipatov (BFKL) equation in N[Formula: see text]=[Formula: see text]4 SYM in terms of the meromorphic functions obtained by the analytic continuation of harmonic sums from positive even integer values of the argument to the complex plane. The meromorphic functions we discuss have pole singularities at negative integers and take finite values at all other points. We derive the reflection identities for harmonic sums at weight five decomposing a product of two harmonic sums with mixed pole structure into a linear combination of terms each having a pole at either negative or non-negative values of the argument. The pole decomposition demonstrates how the product of two simpler harmonic sums can build more complicated harmonic sums at higher weight. We list a minimal irreducible set of bilinear reflection identities at weight five which presents the main result of the paper. We show how the reflection identities can be used to restore the functional form of the next-to-leading eigenvalue of the color-singlet BFKL equation in N[Formula: see text]=4[Formula: see text]SYM, i.e. we argue that it is possible to restore the full functional form on the entire complex plane provided one has information how the function looks like on just two lines on the complex plane. Finally we discuss how nonlinear reflection identities can be constructed from our result with the use of well known quasi-shuffle relations for harmonic sums.

Uniqueness of Q-Difference Polynomials of Meromorphic Functions

2013 ◽  
Vol 756-759 ◽  
pp. 2948-2951 ◽  
Author(s):  
Ke Yu Zhang
Keyword(s):  
Meromorphic Functions ◽  
Uniqueness Problem ◽  
Uniqueness Theorems ◽  
Difference Polynomial ◽  
Difference Polynomials

In this paper, Applying the theory of Nevanlinna, we investigated uniqueness problem of difference polynomial of meromorphic functions and obtained uniqueness theorems of meromorphic functions , which Extended and improved the results of literature [5].

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