Paley Effect for Entire Dirichlet Series

Hlova, T. Ya.; Filevych, P. V.

doi:10.1007/s11253-015-1117-x

Ukr Math J

2015

DOI: 10.1007/s11253-015-1117-x

|View full text |Cite

Paley Effect for Entire Dirichlet Series

T. Ya. Hlova¹,

P. V. Filevych

Abstract: UDC 517.53For the entire Dirichlet series f (z) = X 1 n=0 ane zλn , we establish necessary and sufficient conditions on the coefficients an and exponents λn under which the function f has the Paley effect, i.e., the condition lim supis satisfied, where M f (r) and T f (r) are the maximum modulus and the Nevanlinna characteristic of the function f, respectively.

Help me understand this report

Search citation statements

Order By: Relevance

Paper Sections

Select...

Citation Types

Supporting

Mentioning

Contrasting

Year Published

2023

Publication Types

Select...

Article1

Relationship

Self Cite1

Independent0

Authors

Journals

Cited by 1 publication

References 6 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

On regular variation of entire Dirichlet series

Filevych

Hrybel

2023

Mat. Stud.

Self Cite

View full text Add to dashboard Cite

Consider an entire (absolutely convergent in $\mathbb{C}$) Dirichlet series $F$ with the exponents $\lambda_n$, i.e., of the form $F(s)=\sum_{n=0}^\infty a_ne^{s\lambda_n}$, and, for all $\sigma\in\mathbb{R}$, put $\mu(\sigma,F)=\max\{|a_n|e^{\sigma\lambda_n}:n\ge0\}$ and $M(\sigma,F)=\sup\{|F(s)|:\operatorname{Re}s=\sigma\}$. Previously, the first of the authors and M.M.~Sheremeta proved that if $\omega(\lambda)<C(\rho)$, then the regular variation of the function $\ln\mu(\sigma,F)$ with index $\rho$ implies the regular variation of the function $\ln M(\sigma,F)$ with index $\rho$, and constructed examples of entire Dirichlet series $F$, for which $\ln\mu(\sigma,F)$ is a regularly varying function with index $\rho$, and $\ln M(\sigma,F)$ is not a regularly varying function with index $\rho$. For the exponents of the constructed series we have $\lambda_n=\ln\ln n$ for all $n\ge n_0$ in the case $\rho=1$, and $\lambda_n\sim(\ln n)^{(\rho-1)/\rho}$ as $n\to\infty$ in the case $\rho>1$. In the present article we prove that the exponents of entire Dirichlet series with the same property can form an arbitrary sequence $\lambda=(\lambda_n)_{n=0}^\infty$ not satisfying $\omega(\lambda)<C(\rho)$. More precisely, if $\omega(\lambda)\ge C(\rho)$, then there exists a regularly varying function $\Phi(\sigma)$ with index $\rho$ such that, for an arbitrary positive function $l(\sigma)$ on $[a,+\infty)$, there exists an entire Dirichlet series $F$ with the exponents $\lambda_n$, for which $\ln \mu(\sigma, F)\sim\Phi(\sigma)$ as $\sigma\to+\infty$ and $M(\sigma,F)\ge l(\sigma)$ for all $\sigma\ge\sigma_0$.

show abstract

On regular variation of entire Dirichlet series

Filevych

Hrybel

2023

Mat. Stud.

Self Cite

View full text Add to dashboard Cite

show abstract

scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.

Contact Info

customersupport@researchsolutions.com

10624 S. Eastern Ave., Ste. A-614

Henderson, NV 89052, USA

Blog Terms and Conditions API Terms Privacy Policy Contact Cookie Preferences Do Not Sell or Share My Personal Information

Made with 💙 for researchers

Part of the Research Solutions Family.

Paley Effect for Entire Dirichlet Series

Cited by 1 publication

References 6 publications

On regular variation of entire Dirichlet series

On regular variation of entire Dirichlet series

Contact Info

Product

Resources

About