# On regular variation of entire Dirichlet series

• P. V. Filevych Lviv Polytechnic National University
• O. B. Hrybel Department of Mathematics, Lviv Polytechnic National University, Lviv, Ukraine
Keywords: slowly varying function; regularly varying function; Dirichlet series; supremum modulus; maximal term; central index; Young-conjugate function

### Abstract

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 $\sigma\ge\sigma_0$.

### Author Biographies

P. V. Filevych, Lviv Polytechnic National University

Department of Mathematics, Lviv Polytechnic National University
Lviv, Ukraine

O. B. Hrybel, Department of Mathematics, Lviv Polytechnic National University, Lviv, Ukraine

School of Mathematics, University of Bristol,
Bristol, United Kingdom,
Faculty of Mathematics and Computer Science, Vasyl Stefanyk Precarpathian National University,
Ivano-Frankivsk, Ukraine

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Published
2023-01-16
How to Cite
Filevych, P. V., & Hrybel, O. B. (2023). On regular variation of entire Dirichlet series. Matematychni Studii, 58(2), 174-181. https://doi.org/10.30970/ms.58.2.174-181
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Articles