  We prove that the norm of the Riesz projection from $L^\infty(\Bbb{T}^n)$ to
$L^p(\Bbb{T}^n)$ 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
$H^p(\Bbb{T}^{\infty})$ does not contain the dual space of
$H^1(\Bbb{T}^{\infty})$ for any $p>2$. We then note that the dual of
$H^1(\Bbb{T}^{\infty})$ 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 $L^p$ norm when
$1<p<\infty$, and we use this result to show that the $L^\infty$ norm of the
$N$th partial sum of a bounded Dirichlet series over $d$-smooth numbers is of
order $\log\log N$.
