Optimal comparison of $P$-norms of Dirichlet Polynomials

Let $1\le p<q<\mathrm{\infty }$. We show that $$\sup \frac{{\parallel D\parallel }_{{\mathcal{H}}_q}}{{\parallel D\parallel }_{{\mathcal{H}}_p}}=\exp \left(\frac{\log x}{\log \log x}(\log \sqrt{\frac{q}{p}}+\left(\frac{\log \log \log x}{\log \log x}\right))\right),$$ where the supremum is taken over all non-zero Dirichlet polynomials of the form $D(s)={\sum }_{n\le x}{a}_n{n}^{-s}$. An aplication is given to the study of multipliers between Hardy spaces of Dirichlet series.