Random Dirichlet series with α-stable coefficients

Let P be a set of non-negative real numbers, we consider a class of random Dirichlet series D(s)=∑p∈Pξpps with symmetric α-stable (0<α⩽2) coefficients. Real zero point problem of D is studied firstly. We prove that, if Z(τz)<∞, then with a positive probability there are no real zero points in the interval [τz/α,∞), while if Z(τz)=∞, for any ɛ>0, almost surely D has infinite number of real zeros in the interval s=(τz/α,τz/α+ɛ). Here, Z(s)=∑p∈P1ps and τz is its abscissa of convergence (see Section 1). In the proof, we find that lim sups→1/α+D(s)=∞ almost surely under the condition Z(τz)=∞. Finally, to get more asymptotic information for D(s), under P=N, some asymptotic properties for D(s) compared with Z(αs) are explored as s→1/α+.