We study the supremum of some random Dirichlet polynomials D _N(t) = Σ ^N _n=2=2 ε _nd _nn- ^σ-it, where (ε _n) is a sequence of independent Rademacher random variables, the weights (d _n) are multiplicative and 0 ≤ σ < 1/2. Particular attention is given to the polynomials Σ _n∈ετ = {2 ≤ n ≤ N:P ⁺(n) ≤ p _τ}, P ⁺(n) being the largest prime divisor of n. We obtain sharp upper and lower bounds for the supremum expectation that extend the optimal estimate of Halász-Queffélec, double-script E sign sup _t∈ℝ|Σ ^N _n=2 εn ^n-σ-it| ≈ N ^1-σ/ _logN The proofs are entirely based on methods of stochastic processes, in particular the metric entropy method.