Optimal comparison of $P$-norms of Dirichlet Polynomials

TL;DR

This paper derives an optimal asymptotic expression for the ratio of p-norms of Dirichlet polynomials, revealing precise growth behavior and implications for Hardy space multipliers.

Contribution

It provides the first sharp asymptotic formula for the supremum of p-norm to q-norm ratios of Dirichlet polynomials, advancing understanding of their norm behavior.

Findings

Explicit asymptotic formula for the ratio of norms

Identification of growth rate involving logarithmic factors

Application to Hardy space multipliers

Abstract

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