Boundary behaviour of Dirichlet series with applications to universal series (Q2830647)

The authors provide several theorems on the relationship between the boundary behaviour of a general Dirichlet series NEWLINE\[NEWLINE f(s) = \sum_{n=1}^\infty a_n\, e^{-\lambda_n s}, \quad s=\sigma+it \in \mathbb{C}, NEWLINE\]NEWLINE where \((\lambda_n) \subset \mathbb{R}_{\geq 0}\) is unbounded and strictly increasing, and a subesequence \((S_{m_k})\) of the partial sums NEWLINE\[NEWLINE S_m(s) := \sum_{n=1}^m a_n\, e^{-\lambda_n s}. NEWLINE\]NEWLINE Let \(f \in D(\mathbb{C}_+)\), that is the space of all holomorphic functions on \(\{ \sigma > 0 \}\) with a representation as an absolutely convergent Dirichlet series, and NEWLINE\[NEWLINE E := \{ \zeta \in i\mathbb{R} : S(\zeta) := \lim_{k \to \infty} S_{m_k}(\zeta) \text{ exists} \}, \quad F := \{ \zeta \in i\mathbb{R} : f(\zeta) := \text{nt}\lim_{s \to \zeta} f(s) \text{ exists} \}, NEWLINE\]NEWLINE where \(\mathrm{nt}\,\lim\) denotes the (finite) nontangential limit, then \(f=S\) a.\,e. on \(E \cap F\).NEWLINENEWLINEBy a fat approach region to a point \(it_0\) on the imaginary axis, a region NEWLINE\[NEWLINE \Omega(it_0) := \{ \sigma+it : |t-t_0| < a \text{ and } \phi(t-t_0) < \sigma < b \} NEWLINE\]NEWLINE is meant, where \(a, b > 0\) and \(\phi : [-a,a] \to [0,\infty)\) is a Lipschitz function satisfying \(\int_{-a}^a \frac{\phi(y)}{y^2} dy < \infty\), e.g., half-discs or sets of the form \(\{ \sigma+it : |t|^\alpha < \sigma < 1 \}, \alpha > 1\).NEWLINENEWLINELet \(f \in D(\mathbb{C}_+)\) be bounded on two fat approach regions, \(\Omega(it_1)\) and \(\Omega(it_2)\), \(t_1 < t_2\). If \((S_{m_k})\) is uniformly bounded on an open interval of \(i\mathbb{R}\) containing \(it_1, it_2\), then \(f\) is bounded on \((0,1) \times (t_1,t_2)\). Let \(f \in D(\mathbb{C}_+)\), where \(f'\) is bounded on a fat approach region to \(\zeta \in i\mathbb{R}\). If \((S_{m_k})\) is uniformly bounded on an open interval of \(i\mathbb{R}\) containing \(\zeta\), and \(\lambda_{m_k+1}/\lambda_{m_k} \to \infty\), then \(\lim_{k \to \infty} S_{m_k}(\zeta) = \text{nt}\lim_{s \to \zeta} f(s)\). Their interesting results have also applications to universal ordinary (\(\lambda_n = \log n\)) Dirichlet series \(f \in UD_0\) and to universal Taylor series on the unit-disc \(\mathbb{D}\) with respect to the origin, \(f \in U(\mathbb{D},0)\), respectively.NEWLINENEWLINELet \(f \in UD_0\). Then for a.\,e. \(\zeta \in i\mathbb{R}\) the set \(f(\Gamma)\) is dense in \(\mathbb{C}\) for every open isosceles triangle \(\Gamma \subset \mathbb{C}_+\) with vertex at \(\zeta\) and horizontal axis of symmetry. And, \(f\) cannot be bounded on fat approach regions to two different points. Let \(f \in U(\mathbb{D},0)\). Then \(f'\) cannot be bounded on any fat approach region to a point of the boundary of \(\mathbb{D}\).