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 C) Dirichlet series F with the exponents λn, i.e., of the form F(s)=n=0anesλn, and, for all σR, put μ(σ,F)=max 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

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.

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
Section
Articles