Optimal comparison of $P$-norms of Dirichlet Polynomials

Abstract

Let $1\le p<q<\infty$. 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 $D(s)={\sum }_{n\le x}{a}_n{n}^{-s}$. An aplication is given to the study of multipliers between Hardy spaces of Dirichlet series.

11 pages