Expected Number of Real Zeros of a Random Polynomial With Independent Identically Distributed Symmetric Long-Tailed Coefficients

We show that the expected number of real zeros of the nth degree polynomial with real independent identically distributed coefficients with common characteristic function φ(z) = e-A(ln|1/z|)^-a for 0 < |z| < 1 and φ(0) = 1, φ(z) ≡ 0 for 1 ≦ |z| < ∞, with 1 < a and A ≧ a(a-1), is (a-1)/(a-1/2) log(n) asymptotically as n → ∞.