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We show that \\[ \\sup{\\frac{\\left\\| D\\right\\|_{\\mathcal{H}_{q}}}{\\left\\| D\\right\\|_{\\mathcal{H}_{p}}}} = \\exp{\\left( \\frac{\\log{x}}{\\log{\\log{x}}} \\left(\\log{\\sqrt{\\frac{q}{p}}} + \\left(\\frac{\\log{\\log{\\log{x}}}}{\\log{\\log{x}}}\\right)\\right) \\right)} \\,,\\] where the supremum is taken over all non-zero Dirichlet polynomials of the form $D(s)=\\sum_{n \\leq x}{a_{n} n^{-s}}$. 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