Multipliers of Dirichlet series and monomial series expansions of holomorphic functions in infinitely many variables

Let ${\mathcal{H}}_{\mathrm{\infty }}$ be the set of all ordinary Dirichlet series $D={\sum }_n{a}_n{n}^{-s}$ representing bounded holomorphic functions on the right half plane. A multiplicative sequence $(b_n)$ of complex numbers is said to be an ${\mathrm{\ell }}_1$-multiplier for ${\mathcal{H}}_{\mathrm{\infty }}$ whenever $\sum_n |a_n b_n| < \infty$forevery$D \in $\mathcal{H}$\infty$. We study the problem of describing such sequences $(b_n)$ in terms of the asymptotic decay of the subsequence $(b_{p_j})$, where $p_j$ denotes the $j$th prime number. Given a multiplicative sequence $b=(b_n)$ we prove (among other results): $b$ is an ${\mathrm{\ell }}_1$-multiplier for ${\mathcal{H}}_{\mathrm{\infty }}$ provided $\mathrm{\mid }{b}_{p_j}\mathrm{\mid }<1$ for all $j$ and ${\stackrel{}{\lim }}_n\frac{1}{\log n}{\sum }_{j=1}^n{b}_{p_j}^{\ast 2}<1$, and conversely, if $b$ is an ${\mathrm{\ell }}_1$-multiplier for ${\mathcal{H}}_{\mathrm{\infty }}$, then $\mathrm{\mid }{b}_{p_j}\mathrm{\mid }<1$ for all $j$ and $\overline{\lim}n \frac{1}{\log n} \sum{j=1}^n b{p_j}^{2} \leq 1$(here$b^$ stands for the decreasing rearrangement of $b$). Following an ingenious idea of Harald Bohr it turns out that this problem is intimately related with the question of characterizing those sequences $z$ in the infinite dimensional polydisk ${\mathbb{D}}^{\mathrm{\infty }}$ (the open unit ball of ${\mathrm{\ell }}_{\mathrm{\infty }}$) for which every bounded and holomorphic function $f$ on ${\mathbb{D}}^{\mathrm{\infty }}$ has an absolutely convergent monomial series expansion ${\sum }_{\alpha }\frac{{\mathrm{\partial }}_{\alpha }f(0)}{\alpha !}{z}^{\alpha }$. Moreover, we study analogous problems in Hardy spaces of Dirichlet series and Hardy spaces of functions on the infinite dimensional polytorus ${\mathbb{T}}^{\mathrm{\infty }}$.