Analytic continuation of random Dirichlet series
Let \(\Omega = (\Omega, A, P)\) be a probability space. If \(m_1, \dots, m_r \in \mathbb N\), let \(\varepsilon_{m_1, \dots, m_r}\) be independent random variables defined on \(\Omega\) which take only the values \(-1\) and \(+1\) with equal probability (=1/2). To such a sequence one can associate a multiple Dirichlet series \[ F_r(s, \omega):= \sum_{m_1=1}^\infty \dots \sum_{m_r = 1}^\infty \frac{\varepsilon_{m_1, \dots, m_r}(\omega)X(m_1, \dots, m_r)}{(m_1 + \dots +m_r)^s} \] where \(s \in \mathbb C\) and \(X(m_1, \dots, m_r) \in \mathbb R\). \(F_r(s, \omega)\) can be written as a single Dirichlet series \[ F_r(s, \omega) = \sum_{n=1}^\infty Y_r(n, \omega)n^{-s} \] where \[ Y_r(n, \omega) = \sum_{m_1+ \dots + m_r = n}\varepsilon_{m_1, \dots, m_r}(\omega)X(m_1, \dots, m_r) \] is a series which, for any fixed \(\omega \in \Omega\), has an abscissa of convergence \(\sigma_c(F_r, \omega)\). According to the zero-one law, there is a constant \(\sigma_c(F_r)\) such that \(\sigma_c(F_r, \omega) = \sigma_c(F_r)\) almost certainly. The first result states that the line Re \((s) = \sigma_c(F_r)\) is the natural boundary of \(F_r(s, \omega)\) almost certainly. The second results deals with the case when \(X(m_1, \dots, m_r)\) is the product \(\Lambda(m_1) \dots \Lambda(m_r)\) of the von Mangoldt functions. Then the corresponding random series converges for Re \((s) > r/2\) almost certainly.
scientific article; zbMATH DE number 6380772
scientific article