Real zeros of random Dirichlet series

Let F(sigma) be the random Dirichlet series F(sigma) = Sigma(p is an element of P) X-p/p(sigma), where P is an increasing sequence of positive real numbers and (X (p) ) (p is an element of P) is a sequence of i.i.d. random variables with IP(X-1 = 1) = IP(X-1 = -1) = 1/2. We prove that, for certain conditions on P, if Sigma(p is an element of P) 1/p < infinity then with positive probability F(sigma) has no real zeros while if Sigma(p is an element of P) 1/p = infinity, almost surely F(sigma) has an infinite number of real zeros.