Riesz projection and bounded mean oscillation for Dirichlet series

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