On the maximum of the modulus and the maximal term of Dirichlet series
For \(\varLambda=(\lambda_n)_{n=0}^\infty\subset[0,+\infty)\) with \(\lambda_0=0\), \(\lambda_n\nearrow+\infty\), and \(A\in(-\infty,+\infty]\), let \(S(\varLambda,A)\) be the space of all Dirichlet series \(F(s)=\sum_{n=0}^\infty a_ne^{s\lambda_n}\) for which \(A\) is the abscissa of absolute convergence. Put \(n(t):=\sum_{\lambda_n\leq t}1\), \(M(\sigma,F):=\sup\{| F(\sigma+it)| : t\in\mathbb R\}\), \(\mu(\sigma, F):=\max\{| a_n| e^{\sigma\lambda_n}: n\geq0\}\). Let \(\varOmega(A)\) denote the set of all unbounded \(\mathcal C^2\)--functions \(\varPhi:(-\infty,A)\longrightarrow(0,+\infty)\) such that \(\varPhi'(\sigma)>0\), \(\varPhi''(\sigma)\geq0\), \(\sigma\in(-\infty,A)\), and \(\varPhi'(\sigma)\nearrow+\infty\) when \(\sigma\nearrow A\). The main result of the paper is the following theorem. Assume that \(A\in\{0,+\infty\}\), \(\varPhi\in\varOmega(A)\) are such that \(\varPhi'(\sigma)/\varPhi(\sigma)\nearrow+\infty\) and \(\log\varPhi'(\sigma)=o(\varPhi(\sigma))\) when \(\sigma\nearrow A\). Let \(\varphi\) be the inverse function to \(\varPhi'\), \(\varPsi(\sigma):=\sigma-\varPhi(\sigma)/\varPhi'(\sigma)\). Then \(\Big(\forall_{F\in S(\varLambda,A)}: (\log\mu(\sigma,F)\leq(1+o(1))\varPhi(\sigma),\;\sigma\nearrow A) \Longleftrightarrow (\log M(\sigma,F)\leq(1+o(1))\varPhi(\sigma),\;\sigma\nearrow A)\Big) \Longleftrightarrow\log n(t)=o(\varPhi(\varPsi(\varphi(t)))),\;t\to+\infty\).
scientific article; zbMATH DE number 2115579