关于Dirichlet级数在半平面上绝对收敛的极大项的性质的注记

Q3 Mathematics Matematychni Studii Pub Date : 2021-12-27 DOI:10.30970/ms.56.2.144-148
M. Sheremeta
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引用次数: 0

摘要

通过$S_0(\Lambda)$表示一类狄利克雷级数$F(S)=\sum_{n=0}^{\infty}a_n\exp\{S\Lambda\}(S=\sigma+it)$,并增加到指数($\Lambda_0=0$)的$+\infty$序列$\Lambda=(\Lambda_n)$和绝对收敛的横坐标$\sigma_a=0$。\ln\Lambda_n=o(\ln|a_n|)$$(n\to\infty)$。设$\mu(\sigma,F)=\max\{|a_n|\exp{(\sigma_labda\n)}\colonn\ge0\}$为Dirichlet级数的最大项。证明了对于S_0^*(\Lambda)$中的每一个函数$F\,$\ln(1/|\sigma|)=o(\ln\mu(\sigma\uparrow0)$,$\displaystyle\varlimsup\limits_to\infty}\frac{\ln\Lambda_{n+1}}{\ln\Lambda_n}<+\infty是充要条件$对于磁盘$\{z\colon|z|<1\}$函数$f(z)=\sum_{n=0}^{\fty}a_n z^n$和$r\in(0,1)$中的分析,我们放入$M_f(r)=\max\{|f(z,|\colon|z |=r<1\}$和$\mu_f(r)=\max\。因此,我们得到以下语句:{\sl如果存在一个序列$(n_j)$,使得$\ln n_{j+1}=O(\ln n_{j})$和$\ln n_{j}=O(\ln|a_{n_{j}}|)$为$j\to\infty$,则函数$\ln \mu_f(r)$和$\ln M_f(r)美元是否同时缓慢增加。
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Note to the behavior of the maximal term of Dirichlet series absolutely convergent in half-plane
By $S_0(\Lambda)$ denote a class of Dirichlet series $F(s)=\sum_{n=0}^{\infty}a_n\exp\{s\lambda_n\} (s=\sigma+it)$ withan increasing to $+\infty$ sequence $\Lambda=(\lambda_n)$ of exponents ($\lambda_0=0$) and the abscissa of absolute convergence $\sigma_a=0$.We say that $F\in S_0^*(\Lambda)$ if $F\in S_0(\Lambda)$ and $\ln \lambda_n=o(\ln |a_n|)$ $(n\to\infty)$. Let$\mu(\sigma,F)=\max\{|a_n|\exp{(\sigma\lambda_n)}\colon n\ge 0\}$ be the maximal term of Dirichlet series. It is proved that in order that $\ln (1/|\sigma|)=o(\ln \mu(\sigma))$ $(\sigma\uparrow 0)$ for every function $F\in S_0^*(\Lambda)$ it is necessary and sufficient that $\displaystyle \varlimsup\limits_{n\to\infty}\frac{\ln \lambda_{n+1}}{\ln \lambda_n}<+\infty. $For an analytic in the disk $\{z\colon |z|<1\}$ function $f(z)=\sum_{n=0}^{\infty}a_n z^n$ and $r\in (0, 1)$ we put $M_f(r)=\max\{|f(z)|\colon |z|=r<1\}$ and $\mu_f(r)=\max\{|a_n|r^n\colon n\ge 0\}$. Then from hence we get the following statement: {\sl if there exists a sequence $(n_j)$ such that $\ln n_{j+1}=O(\ln n_{j})$ and $\ln n_{j}=o(\ln |a_{n_{j}}|)$ as $j\to\infty$,  then the functions $\ln \mu_f(r)$ and $\ln M_f(r)$ are or not are slowly increasing simultaneously.
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来源期刊
Matematychni Studii
Matematychni Studii Mathematics-Mathematics (all)
CiteScore
1.00
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0.00%
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38
期刊介绍: Journal is devoted to research in all fields of mathematics.
期刊最新文献
On the h-measure of an exceptional set in Fenton-type theorem for Taylor-Dirichlet series Almost periodic distributions and crystalline measures Reflectionless Schrodinger operators and Marchenko parametrization Existence of basic solutions of first order linear homogeneous set-valued differential equations Real univariate polynomials with given signs of coefficients and simple real roots
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