Convergence almost everywhere and divergence everywhere of Taylor and Dirichlet series (Q1769691)

Nine theorems are proved. The following ones are typical. Theorem 3.2. There exists a Dirichlet series \[ f(x):= \sum^\infty_{n=1} a_n n^{-s} \] with convergence and boundedness in the half-plane \(\mathbb{C}_0:=\{s\in\mathbb{C}:\text{Re}(s)> 0\}\), and such that \(\sum a_n n^{it}\) diverges for each \(t\in\mathbb{R}\). Theorem 5.3. Let \((a_n\geq 1)\) be a sequence of complex numbers such that \(|a_n|\) decreases to zero. Then (i) There exists a sequence \((\theta_n: n\geq 1)\) of complex signs \((|a_n|= 1)\) such that \(\sum\theta_n a_n n^{it}\) converges for each \(t\in\mathbb{R}\). (ii) In case \(\sum|a_n|/n<\infty\), one can choose the signs \(\theta_n\) to be real \((\theta_n= \pm1)\) in statement (i).