# On regular variation of entire Dirichlet series

### 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 $\sigma\ge\sigma_0$.

### References

E. Seneta, Regularly Varying Functions, Springer-Verlag, Berlin, Heidelberg, New York (1976).

M.V. Zabolotskyi, M.M. Sheremeta, On the slow growth of the main characteristics of entire functions, Math. Notes, 65 (1999), №2, 168-174. https://doi.org/10.1007/BF02679813

P.V. Filevych, M.M. Sheremeta, On the regular variation of main characteristics of an entire function, Ukr. Math. J., 55 (2003), №6, 1012-1024. https://doi.org/10.1023/B:UKMA.0000010600.46493.2c

P.V. Filevych, M.M. Sheremeta, Regularly increasing entire Dirichlet series, Math. Notes, 74 (2003), №1, 110-122. https://doi.org/10.1023/A:1025027418525

M.M. Dolynyuk, O.B. Skaskiv, About the regular growth of some positive functional series, Nauk. Visn. Cherniv. National Univer. Mat., (2006), Iss. 314–315, 50–58. https://bmj.fmi.org.ua/index.php/adm/article/view/519

T.Ya. Hlova, P.V. Filevych, Paley effect for entire Dirichlet series, Ukr. Math. J., 67 (2015), №6, 838-852. https://doi.org/10.1007/s11253-015-1117-x

T.Ya. Hlova, P.V. Filevych, Generalized types of the growth of Dirichlet series, Carpathian Math. Publ., 7 (2015), №2, 172-187. https://doi.org/10.15330/cmp.7.2.172-187

T.Ya. Hlova, P.V. Filevych, The growth of entire Dirichlet series in terms of generalized orders, Sb. Math., 209 (2018), №2, 241-257. https://doi.org/10.1070/SM8644

M.M. Sheremeta, On the growth of an entire Dirichlet series, Ukr. Math. J., 51 (1999), №8, 1296-1302. https://doi.org/10.1007/BF02592520

P.V. Filevych, On Valiron's theorem on the relations between the maximum modulus and the maximal term of an entire Dirichlet series, Russ. Math., 48 (2004), №4, 63-69.

*Matematychni Studii*,

*58*(2), 174-181. https://doi.org/10.30970/ms.58.2.174-181

Copyright (c) 2023 O. Hrybel, P. Filevych

This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Matematychni Studii is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International (CC BY-NC-ND 4.0) license.