On regular variation of entire Dirichlet series
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 σ∈R, 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.
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