Natural boundary of Dirichlet series
Gautami Bhowmik, Jan-Christoph Schlage-Puchta
证明了狄利克雷级数自然边界存在的一些条件。我们表明，一般假定的边界是自然的。我们还给出了自然边界在确定渐近结果中的应用。
We prove some conditions on the existence of natural boundaries of Dirichlet series. We show that generically the presumed boundary is the natural one. We also give an application of natural boundaries in determining asymptotic results.
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Construction of an Ordinary Dirichlet Series with Convergence beyond the Bohr Strip
一个普通的狄利克雷级数有三个横坐标，分别描述了狄利克雷级数收敛、一致收敛和绝对收敛的最大区域。1931年，Hille和Bohnenblust关于狄利克雷级数可以一致收敛但不是绝对收敛的区域的论文，促使人们对这个区域进行了大量的研究，即"玻尔带"。然而，一个相关的自然问题显然没有得到回答：对于一个具有非平凡玻尔带的狄利克雷级数，级数在玻尔带之外多远收敛？我们研究这个问题的明确建设，创造狄利克雷级数收敛超出他们的玻尔带。
We prove an analogue of the Lagrange Inversion Theorem for Dirichlet series. The proof is based on studying properties of Dirichlet convolution polynomials, which are analogues of convolution polynomials introduced by Knuth in [4].
One metric result about analytic continuation of some Dirichlet series
本文考虑一类单参数狄利克雷级数族。对于一个特定的参数值的系列变成黎曼ζ函数的狄利克雷级数。我们证明了该族的几乎每个级数都具有到半平面Re s > 1/2的解析延拓，其中它不消失。这一结果是由不同的作者以前得到的。我们给出了它的简单证明的一些三角和的估计。
On universality of general Dirichlet series
本文给出一般频率狄利克雷级数在垂直平移下普适的充分条件，可应用于赫尔维茨ζ函数等，并新提出交替素数ζ函数作为普适例子。
Estimates for the number of zeros of shifted combinations of completed Dirichlet series
本文为完备 Dirichlet 级数平移组合的一个子类在临界线上的零点数量提供了下界估计。
This note deals with the relationship between the abscissas of simple, uniform and absolute convergence for the Dirichlet series $f(s) = \sum_{n=1}^\infty a_n n^{-s}$, when the coefficients $a_n$ are either multiplicative or completely multiplicative.
This paper establishes connections between the boundary behaviour of functions representable as absolutely convergent Dirichlet series in a half-plane and the convergence properties of partial sums of the Dirichlet series on the boundary. This yields insights into the boundary behaviour of Dirichlet series and Taylor series which have universal approximation properties.
On generalized Hardy classes of Dirichlet series
本文将Hedenmalm，Lindqvist，Olofsson，Olsen，Saksman，Seip等人研究的狄利克雷级数的Hardy类H^2推广到更一般的狄利克雷级数.我们证明了这类函数的一些结果，例如在短区间上对数L^1-模的估计。我们把这一点联系起来，并利用这些结果使我们的狄利克雷级数的一个最近的非零结果更加明确。特别是，我们给一个应用程序的赫维茨zeta函数。
We characterize the uniform limits of Dirichlet polynomials on a right half plane. In the Dirichlet setting, we find approximation results, with respect to the Euclidean distance and {to} the chordal one as well, analogous to classical results of Runge, Mergelyan and Vitushkin. We also strengthen the notion of universal Dirichlet series.
It is shown that the algebra of bounded Dirichlet series is not a coherent ring, and has infinite Bass stable rank. As corollaries of the latter result, it is derived that the algebra of bounded Dirichlet series has infinite topological stable rank and infinite Krull dimension.
Fundamental domains are found for functions defined by general Dirichlet series and using basic properties of conformal mappings the Great Riemann Hypothesis is studied.
This paper investigates lower bounds on the number of zeros and poles of a general Dirichlet series in a disk of radius $r$ and gives, as a consequence, an affirmative answer to an open problem of Bombieri and Perelli on the bound. Applications will also be given to Picard type theorems, global estimates on the symmetric difference of zeros, and uniqueness problems for Dirichlet series.
We establish necessary and sufficient conditions for boundedness of composition operators on the most general class of Hilbert spaces of entire Dirichlet series with real frequencies. Depending on whether or not the space contains any nonzero constant function, different criteria for boundedness are developed. Thus, we complete the characterization of bounded composition operators on all known Hilbert spaces of entire Dirichlet series of one variable.
For the Dirichlet series of the form $\displaystyle F(z,ω)=\sum\nolimits_{k=0}^{+\infty} f_k(ω)e^{zλ_k(ω)} $ $ (z\in\mathbb{C},$ $ω\inΩ)$ with pairwise independent real exponents $(λ_k(ω))$ on probability space $(Ω,\mathcal{A},P)$ an estimates of abscissas convergence and absolutely convergence are established.
Zeros of functions in Bergman-type Hilbert spaces of Dirichlet Series
对于实数$α$，希尔伯特空间$\mathscr{D}_α$由那些狄利克雷级数$\sum_{n=1}^\infty a_n/n^s$组成，其中$\sum_{n=1}^\infty |a_n|^2/[d(n)]^α< \infty$，其中$d(n)$表示$n$的约数。本文将Seip关于$\mathscr{D}_α$中函数的有界零点列的一个定理推广到$α>0$中.本文还讨论了狄利克雷级数在其它加权空间中的推广，以及$1\leq p <2$在狄利克雷级数$\mathscr{H}^p$的Hardy空间中函数零点的部分结果.
In this paper we wish to study some growth properties of entire functions represented by a vector valued Dirichlet series on the basis of (p,q)th relative Ritt order, (p,q)th relative Ritt type and (p,q)th relative Ritt weak type.
Representation of Infinitely Differentiable Functions by Dirichlet Series
研究实直线上无穷可微函数加权空间元素的指数级数表示问题。
Analytic continuation of Dirichlet series with almost periodic coefficients
证明了当g是奇1周期实解析函数且b是丢番图时，系数为a（n）=g（n b）的普通狄利克雷级数的收敛横坐标为0.我们还证明了如果g是奇数且有界变差，b是有界丢番图型r>1，则收敛横坐标小于或等于1-1/r。利用多对数展开式，证明了当g是奇实解析的且b是丢番图时，普通狄利克雷级数在整个复平面上有一个解析延拓.
Fourier multipliers for Hardy spaces of Dirichlet series
我们得到了Dirichlet-Hardy空间中Fourier乘子的新结果。作为结果，我们建立了一个Littlewood-Paley型不等式，它给出了一个简单的证明，即狄利克雷单项式形成一个Schauder基p>1。
Bohr and Rogosinski abscissas for ordinary Dirichlet series
本文证明了一般狄利克雷级数的Bohr和Rogosinski定理，将右半平面映射到有界凸区域$G\subset \mathbb{C} $上，与域$G$无关.此外，我们还得到了关于这两个类的新的估计。
Some Remarks on the Algebra of Bounded Dirichlet Series
研究了所有有界于右半半平面上的狄利克雷级数的代数。我们考虑类似的电晕定理在此设置，并表明它是假的，即右半平面是不密集的极大理想空间。我们还证明了一些改进的希勒-Bohnenblust定理，其中使用概率和确定性的设备，我们展示了如何进行每个证明。
We dealt in a series of previous publications with some geometric aspects of the mappings by functions obtained as analytic continuations to the whole complex plane of general Dirichlet series. Pictures illustrating those aspects contain a lot of other information which has been waiting for a rigorous proof. Such a task is partially fulfilled in this paper, where we succeeded among other things, to prove a theorem about general Dirichlet series having as corollary the Speiser's theorem. We have also proved that those functions do not possess multiple zeros of order higher than $2$ and the doub…
We characterize the space of multipliers from the Hardy space of Dirichlet series $\mathcal H_p$ into $\mathcal H_q$ for every $1 \leq p,q \leq \infty$. For a fixed Dirichlet series, we also investigate some structural properties of its associated multiplication operator. In particular, we study the norm, the essential norm, and the spectrum for an operator of this kind. We exploit the existing natural identification of spaces of Dirichlet series with spaces of holomorphic functions in infinitely many variables and apply several methods from complex and harmonic analysis to obtain our results.…
Application of Meyer's theorem on quasicrystals to exponential polynomials and Dirichlet series
本文利用 Meyer 准晶定理，给出了绝对收敛 Dirichlet 级数（虚指数、实零点）为有限正弦乘积的充要条件。
Composition operators on weighted Hilbert spaces of Dirichlet series
本文研究了Dirichlet级数加权Hilbert空间上特征为零的复合算子，通过证明加权均值计数函数的存在性并给出变量替换公式，导出了有界性与紧性的必要条件，并在Bergman型空间中利用Schwarz型引理证明了紧性条件也是充分的。
We show that, in every weighted Dirichlet space on the unit disk with superharmonic weight, the Taylor series of a function in the space is $(C,α)$-summable to the function in the norm of the space, provided that $α>1/2$. We further show that the constant $1/2$ is sharp, in marked contrast with the classical case of the disk algebra.
This paper is associated with Nevanlinna class, Dirichlet series and Szegö's problem in infinitely many variables. As we will see, there is a natural connection between these topics. The paper first introduces the Nevanlinna class and the Smirnov class in this context, and generalizes the classical theory in finitely many variables to the infinite-variable setting. These results applied to Szegö's problem on Hardy spaces in infinitely many variables. Moreover, this paper is also devoted to the study of the correspondence between the Nevanlinna functions and Dirichlet series.
Almost periodicity and boundary values of Dirichlet series
该研究利用概周期性建立狄利克雷级数哈代空间的Hardy–Stein恒等式与Littlewood–Paley公式对应形式，证明相关极限不可交换，揭示两类边值的性质，并探究其零点集与Blaschke乘积。
Integral means and boundary limits of Dirichlet series
本文研究了函数在Hardy空间HD^p中的边界行为，给出了一般狄利克雷级数的边界行为.我们的主要结果回答了H. Hedenmalm的研究表明，经典的F.积分平均的Carlson定理不延伸到HD^ \infty中函数的虚轴，即，对于右半平面H^ \infty中的普通狄利克雷级数.我们讨论了HD^p的一个重要的嵌入问题，该问题的解只有当p是偶数时才知道。把HD^p看作无限维多圆盘的Hardy空间，我们也给出了Fatou定理的类似结果。
Bicomplex Schwarz and Dirichlet Boundary Value Problems
本文在复单位圆盘上定义并求解了双复值函数的Schwarz型与Dirichlet型边值问题，拓展了经典边值问题至双复分析框架。
Local interpolation in Hilbert spaces of Dirichlet series
用$\Hp$表示系数为平方可和的普通狄利克雷级数的希尔伯特空间。主要结果是：半平面$σ>1/2$中的有界点列是$\Hp$的插值列当且仅当它是同一半平面的Hardy空间$H^2$的插值列.对于与半平面$σ>1/2$的Bergman空间和Dirichlet空间相关的普通狄利克雷级数的希尔伯特空间，也得到了类似的局部结果.
We introduce a mean counting function for Dirichlet series, which plays the same role in the function theory of Hardy spaces of Dirichlet series as the Nevanlinna counting function does in the classical theory. The existence of the mean counting function is related to Jessen and Tornehave's resolution of the Lagrange mean motion problem. We use the mean counting function to describe all compact composition operators with Dirichlet series symbols on the Hardy--Hilbert space of Dirichlet series, thus resolving a problem which has been open since the bounded composition operators were described b…
We solve the classical Dirichlet problem for a general complex Hessian equation on a small ball in $\bC^n$. Then, we show that there is a continuous solution, in pluripotential theory sense, to the Dirichlet problem on compact Hermitian manifolds with boundary that equipped locally conformal Kähler metrics, provided a subsolution.
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