Joint value-distribution of Dirichlet series associated with periodic arithmetical functions
Given a positive integer \(q\), an arithmetical function \(f\) is called \(q\)-\textit{periodic} if \(f(n+q) = f(n)\) for all \(n\). Two \(q\)-periodic functions are called (Dirichlet) convolution dependent, if there exist arithmetical functions \(\eta_1, \eta_2\) such that \(f_1 * \eta_1 = f_2 * \eta_2\). Also, a compact set \(K\) inside the strip \(1/2 < \Re s < 1\), with connected complement, is said to be \textit{admissible}. Theorem 1. Let \(f_1\) and \(f_2\) be \(q\)-periodic arithmetical functions and suppose that neither \(f_1\) nor \(f_2\) is vanishing identically and that they are convolution independent. Moreover, let \(K\) be an admissible set and let \(g_1, g_2: K \to \mathbb{C}\) be non-vanishing continuous functions, analytic on the interior of \(K\). Then there exists an admissible subset \(K' \subset K\) containing an open disk such that for every \(\varepsilon > 0\) there exists a positive real number \(\tau\) satisfying \[\max_{j = 1, 2}\max_{s \in K'}\vert L(s + i\tau ; f_j) - g_j(s)\vert < \varepsilon;\] moreover, the set of such shifts \(\tau\) has positive lower density. In the generic case, \(K'\) depends only on \(f_1\) and \(f_2\). Theorem 2. Given two Dirichlet series \(L(s; f_1)\) and \(L(s; f_2)\) associated with \(q\)-periodic functions \(f_1\) and \(f_2\), their meromorphic continuations share the value \(\infty\) if and only if \[\left(\sum_{a\ \mathrm{mod}\ q}f_1(a)\right )\left(\sum_{a\ \mathrm{mod}\ q}f_2(a)\right ) \not = 0.\] Moreover, if a complex value \(c\) is shared, then \(c=0\) and \(f_1 = \lambda f_2\), for some complex number \(\lambda \in \mathbb{C}^*\).
scientific article; zbMATH DE number 7225499
scientific article