On the expected number of roots of a random Dirichlet polynomial

Let T>0 and consider the random Dirichlet polynomial ST(t)=Re∑n≤TXnn−1/2−it, where (Xn)n are i.i.d. Gaussian random variables with mean 0 and variance 1. We prove that the expected number of roots of ST(t) in the dyadic interval [T,2T], say EN(T), is approximately 2/3√ times the number of zeros of the Riemann ζ function in the critical strip up to height T. Moreover, we also compute the expected number of zeros in the same dyadic interval of the kth derivative of ST(t). Our proof requires the best upper bounds for the Riemann ζ function known up to date, and also estimates for the L2 averages of certain Dirichlet polynomials.