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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 aproblem 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 Dirichletseries in $\operatorname{BMOA}$ of the right half-plane. We give severalconditions showing how this $\operatorname{BMOA}$ space relates to other spacesof Dirichlet series. Finally, relating the partial sum operator for Dirichletseries 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 oforder $\log\log N$.

Riesz projection and bounded mean oscillation for Dirichlet series