Convergence almost everywhere and divergence everywhere of Taylor and Dirichlet series.

Abstract

Recent results concerning the convergence almost everywhere or divergence everywhere of Dirichlet series $\sum a_nn^{it}$ appeared in the literature, revealing significant differences with the case of trigonometric series $\sum a_n e^{int}$. In this work, we prove in several cases the optimality of these results. We also discuss the statistical effect of a change of signs, by considering $\sum \pm a_nn^{it}$. According to the way (probabilistic or topological) this change of signs is made, the properties of the resulting series are quite different, and can also be applied to the theory of power series.