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)= Sigma(xi)(P is an element of P)(P)/(S)(P) with symmetric alpha-stable (0 < alpha <= 2) coefficients. Real zero point problem of D is studied firstly. We prove that, if Z(tau(Z))<infinity, then with a positive probability there are no real zero points in the interval [tau(Z)/alpha,infinity), while if Z(tau(Z))=infinity, for any epsilon>0, almost surely D has infinite number of real zeros in the interval epsilon=(tau(Z)/alpha,tau(Z)/alpha+epsilon). Here, Z(S)= Sigma(1)(P is an element of P)/(s)(P) and T-Z is its abscissa of convergence (see Section 1). In the proof, we find that lim sup(s -> 1/alpha+)D(s)=infinity almost surely under the condition Z(tau(Z))=infinity. Finally, to get more asymptotic information for D(s), under P=N, some asymptotic properties for D(s) compared with Z(alpha s) are explored as S -> 1/alpha+.