On general and random Dirichlet series and their partial sums

We consider random Dirichlet series \(f(s)=\sum_{n=1}^{\infty} \varepsilon_n a_n e^{-\lambda_{n} s}\) $f(s)={\sum }_{n=1}^{\infty }{\epsilon }_n{a}_n{e}^{-{\lambda }_{n}s}$ , with \(a_n\) $a_n$ complex numbers, \(\lambda_n \geq 0\) ${\lambda }_n\ge 0$ , increasing to \(\infty\) $\infty$ , and otherwise arbitrary; and with \((\varepsilon_n)\) $({\epsilon }_n)$ a Rademacher sequence of random variables. We study their almost sure convergence on the critical line of convergence \(\{ \text{Re}\,\, s=\sigma_{c}(f)\}.\) $\{\text{Re}s={\sigma }_c(f)\}.$ When \(\lambda_n=n\) ${\lambda }_n=n$ (periodic case), a well-known sufficient condition on the coefficients a_n ensuring almost sure uniform convergence on \([0,2\pi] \) $[0,2\pi ]$ (equivalently uniform convergence on \(\mathbb{R}\) $\mathbb{R}$ ) has been given by Salem and Zygmund, who made strong use of Bernstein's inequality. When \((\lambda_n)\) $({\lambda }_n)$ is arbitrary (non-periodic case), one must distinguish between uniform convergence on compact subsets of \(\mathbb{R}\) $\mathbb{R}$ (local convergence) and uniform convergence on \(\mathbb{R}\) $\mathbb{R}$ . We extend Salem–Zygmund's theorem to general random Dirichlet series in this non-periodic case. Our main tools are a simple “local” Bernstein's inequality, and P. Lévy's symmetry principle.