Today, Hardy spaces of ordinary Dirichlet series \(\sum_n a_n n^{-s}\) play an important role in the modern theory of Dirichlet series. Inspired by ‘Bohr’s vision’, a Dirichlet series \(D(s)=\sum_n a_n n^{-s}\) belongs to the Hardy space \(\mathcal{H}_p, \, 1 \leq p < \infty\), whenever there is \(f \in H_p(\mathbb{T}^\infty)\) (the Hardy space of all \(p\)-integrable \(f\) on the countable product of the circle group \(\mathbb{T}\) with Fourier transforms supported in the positive cone of \(\mathbb{Z}^{(\infty)}\)) such that \(a_n = \widehat{f}(\alpha)\) for every \(n\in \mathbb{N}\) and every finite multi-index \(\alpha = (\alpha_1, \dots , \alpha_N, 0, \dots) \in \mathbb{N}_0^{(\infty)}\) for which \(n = \mathfrak{p}^{\alpha}\) (\(\mathfrak{p}\) the sequence of primes). For \(p=2\) and \(p=\infty\), this definition was initiated by [H. Hedenmalm et al., Duke Math. J. 86, No. 1, 1–37 (1997; Zbl 0887.46008)] and by F. Bayart [Monatsh. Math. 136, No. 3, 203–236 (2002; Zbl 1076.46017)] for arbitrary \(1 \leq p < \infty\) (see also the recent monographs [A. Defant et al., Dirichlet series and holomorphic functions in high dimensions. Cambridge: Cambridge University Press (2019; Zbl 1460.30004)] and [H. Queffélec and M. Queffélec, Diophantine approximation and Dirichlet series. New Delhi: Hindustan Book Agency (2013; Zbl 1317.11001)].
Dirichlet series \(D\) define holomorphic functions on half planes \([\text{Re}(s) > \sigma_c(D)]\), where \(\sigma_c(D) \in \mathbb{R}\) is given by the infimum taken over all \(\text{Re}\, s\) such that \(D\) converges in \(s \in \mathbb{C}\). Defining \(\mathfrak{S}(\mathcal{H}_p) = \sup_{D \in \mathcal{H}_p}\sigma_c(D)\), Bayart proved that \(\mathfrak{S}(\mathcal{H}_p) = \frac{1}{2}\) for \(1 \leq p < \infty\), whereas (by the very definition) \(\mathfrak{S}(\mathcal{H}_\infty) = 0\). A question (going back to H. Hedenmalm) asks for a (hopefully) natural scale of Banach spaces which resolves this jump from \(1 \leq p < \infty\) to \(p = \infty\). More precisely, is there a scale \((X_{\theta})_{ \theta \in [0,1]}\) of Banach spaces which consists of ordinary Dirichlet series and has the property that \(\mathfrak{S}(X_{\theta}) = \theta\) for every \(0 \leq \theta \leq 1\)? This articles gives a positive answer. Defining the Orlicz-Hardy space \(\mathcal{H}_{\Psi_p} : = H_{\Psi_p}(\mathbb{T}^\infty)\) (via Bohr’s vision in analogy to the definition of \(\mathcal{H}_p\)), the main result shows that \(\mathfrak{S}(\mathcal{H}_{\Psi_p}) = \min \{1/p,1/2 \}\), where \(\Psi_p(t) = e^{t^p}-1\) for \(1 \leq p < \infty\) is the well-known exponential Orlicz function.