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Assuming that \\Phi(n) is an increasing function we prove that the set of polynomials {\\sum_{n=2}^N a_n n^{it-1}: N \\geq 2, |a_n| \\leq \\Phi(n)}, is dense in L^2(0,H) if and only if \\sum_{n=2}^\\infty \\frac{\\log \\Phi(n)} {n \\log^2 n} = \\infty. We also prove variants of this result for generalized Dirichlet polynomials. The main tools are theorems of Paley and Wiener related to quasianalyticity and the Pechersky rearrangement theorem. 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