Riesz projection and bounded mean oscillation for Dirichlet series

Abstract: We prove that the norm of the Riesz projection from $L<sup>\infty(\Bbb{T}<sup>n)$ to $L<sup>p(\Bbb{T}<sup>n)$ is $1$ for all $n\ge 1$ only if $p\le 2$, thus solving a problem posed by Marzo and Seip in 2011. This shows that $H<sup>p(\Bbb{T}<sup>{\infty})$ does not contain the dual space of $H<sup>1(\Bbb{T}<sup>{\infty})$ for any $p&gt;2$. We then note that the dual of $H<sup>1(\Bbb{T}<sup>{\infty})$ contains, via the Bohr lift, the space of Dirichlet series in $\operatorname{BMOA}$ of the right half-plane. We give several conditions showing how this $\operatorname{BMOA}$ space relates to other spaces of Dirichlet series. Finally, relating the partial sum operator for Dirichlet series to Riesz projection on $\Bbb{T}$, we compute its $L<sup>p$ norm when $1&lt;p&lt;\infty$, and we use this result to show that the $L<sup>\infty$ norm of the $N$th partial sum of a bounded Dirichlet series over $d$-smooth numbers is of order $\log\log N$.