Regularly Increasing Entire Dirichlet Series

For entire Dirichlet series, we establish conditions on its coefficients and exponents under which the logarithms of the maximal term and of the maximum of the modulus are regularly varying functions of order [1, + ) and the central exponent is a regularly varying function of order – 1.     

On the Stability of the Maximum Term of the Entire Dirichlet Series

We establish necessary and sufficient conditions for the logarithms of the maximum terms of the entire Dirichlet series and to be asymptotically equivalent as Re z → +∞ outside a certain set of finite measure. __________ Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 57, No. 4, pp. 571–576, April, 2005.     

关于随机狄里克莱级数所代表的随机整函数的增长性

Let (Ω,A,P) be a probability spasce. Consider a random Dirichlet series.     

Dirichlet级数表示的整函数的增长性

本文系统地研究了在全平面上收敛的Dirichlet级数的增长性,得到了级数的系数和增长级之间关系的一系列充要条件．     

复指数Dirichlet 级数表示的整函数

该文对一个由复指数Dirichlet 级数表示的，其系数满足一给定增长条件且在一固定水平带形有界不恒为零的整函数的存在性， 给出了充分必要条件。     

缺项广义Dirichlet级数所表示的整函数的增长性

对于有限级广义Dirichlet级数所表示的整函数f(s),引进级ρ_R,型■_R,分别讨论它们与在水平直线上的级ρ_L,型■_L之间的关系．应用于,Taylor级数时,有类似结果．     

一般随机Dirichlet级数所表示的整函数

该文研究了一般随机Dirichlet级数的所表示整函数增长性和值分布，得出了重要结论：在适当条件下，任何水平带形上或水平线上增长级与全面上相同，对于ρ随机Dirichlet级数(0＜ρ＜∞)a.s.在任何宽为〖SX(〗π[]ρ〖SX)〗的水平带形内，至少有一条ρ级没有有穷例外值的Borel线。     

B-值随机Dirichlet级数表示整函数的增长性

系统地研究了全平面收敛的B-值随机Difichlet级数的增长性，得到了在一定条件下，B-值随机Dirichlet级数在收敛平面上的增长（下）级几乎处处等于某Dirichlet级数的增长（下）级还得到了它们与指数和系数的关系式．     

零级缺项广义Dirichlet级数的增长性

在一定的指数条件下,研究了全平面上零级广义Dirichlet级数的增长性,得到了关于它的系数和指数与增长级之间关系的结论.     

On the order of an analytic function represented by Dirichlet Series

The decomposition method is applied to algebraic equations containing exponential terms. The n term approximation φ_n is rapidly damped as n increases, yielding an oscillating convergence of superior accuracy (<0.01%by n = 5 and < 0.0001 % by n = 9 for k = 2).     

On the Mean Values of the Dirichlet Series

For Dirichlet series with arbitrary abscissa of absolute convergence, we investigate the relationhip between the increase in the maximum term and , q ∈ (0,+∞).__________Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 56, No. 11, pp. 1501–1512, November, 2004.     

复指数Dirichlet 级数所表示的整函数的增长性

由Dirichlet 级数表示的整函数f(z)在带形中有界,其系数{a_n}和指数 {λ_n}(n=1,2,... )是复数列.文中引入 φ-级 和下φ -级,讨论f(z) 具有φ -级和下φ -级的条件.     

关于超球级数所定义的整函数

给出了超球级数所定义整函数的下阶、下型与极大项、中心指标、最大模之间的关系及几个不等式，并由此推出了超球级数成正规增长的充要条件及成完全正规增长的充要条件．     

On Dirichlet Series with Periodic Coefficients

We investigate Dirichlet series L(s, f) = _n=1 with q-periodic coefficients f(n), i.e. f(n+q) = f(n) for all integers n and some fixed integer q, and we prove an asymptotic formula for the number of nontrivial zeros of L(s, f). Further, we give a necessary condition for L(s, f) to have a distribution of the nontrivial zeros symmetrical with respect to the critical line.     

On Dirichlet Series for Sums of Squares

Hardy and Wright (An Introduction to the Theory of Numbers, 5th edn., Oxford, 1979) recorded elegant closed forms for the generating functions of the divisor functions _k (n) and _k ²(n) in the terms of Riemann Zeta function (s) only. In this paper, we explore other arithmetical functions enjoying this remarkable property. In Theorem 2.1 below, we are able to generalize the above result and prove that if f _i and g _i are completely multiplicative, then we have where L _f(s) := _{n = 1} f(n)n ^–s is the Dirichlet series corresponding to f. Let r _N(n) be the number of solutions of x ₁ ² + ··· + x _N ² = n and r _2,P (n) be the number of solutions of x ² + Py ² = n. One of the applications of Theorem 2.1 is to obtain closed forms, in terms of (s) and Dirichlet L-functions, for the generating functions of r _N(n), r _N ²(n), r _2,P (n) and r _2,P (n)² for certain N and P. We also use these generating functions to obtain asymptotic estimates of the average values for each function for which we obtain a Dirichlet series.     

On the Approximation of Analytic Functions by Shifts of an Absolutely Convergent Dirichlet Series

Mathematical Notes - A theorem dealing with the approximation of analytic functions in the strip $$\{s\in \mathbb{C}: 1/2&lt; \operatorname{Re} s&lt;1\}$$ by shifts of an absolutely...     

平面上的零级Dirichlet级数

应用最大项指标,在较宽的系数条件下,对复平面上的零级数Dirichlet级数进行了深入的研究,得到关于它们增长性的两个定理,即文中定理1和定理2．     

关于两类狄里克莱级数系数的重排

研究了两类狄里克莱级数的系数重排后的增长性,得到了全平面和半平面上有限级狄里克莱级数的系数经过重排后级和型保持不变的充要条件.