On general and random Dirichlet series and their partial sums

We consider random Dirichlet series \(f(s)=\sum_{n=1}^{\infty} \varepsilon_n a_n e^{-\lambda_{n} s}\), with \(a_n\) complex numbers, \(\lambda_n \geq 0\), increasing to \(\infty\) , and otherwise arbitrary; and with \((\varepsilon_n)\) a Rademacher sequence of random variables. We study their almost sure convergence on the critical line of convergence \(\{ \text{Re}\,\, s=\sigma_{c}(f)\}.\) When \(\lambda_n=n\) (periodic case), a well-known sufficient condition on the coefficients a_n ensuring almost sure uniform convergence on \([0,2\pi] \) (equivalently uniform convergence on \(\mathbb{R}\)) has been given by Salem and Zygmund, who made strong use of Bernstein's inequality. When \((\lambda_n)\) is arbitrary (non-periodic case), one must distinguish between uniform convergence on compact subsets of \(\mathbb{R}\) (local convergence) and uniform convergence on \(\mathbb{R}\). 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.