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Let $\mathcal{H}_\infty$ be the set of all ordinary Dirichlet series$D=\sum_n a_n n^{-s}$ representing bounded holomorphic functions on the righthalf plane. A multiplicative sequence $(b_n)$ of complex numbers is said to bean $\ell_1$-multiplier for $\mathcal{H}_\infty$ whenever $\sum_n |a_n b_n| <\infty$ for every $D \in \mathcal{H}_\infty$. We study the problem ofdescribing 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 $\ell_1$-multiplier for$\mathcal{H}_\infty$ provided $|b_{p_j}| < 1$ for all $j$ and$\overline{\lim}_n \frac{1}{\log n} \sum_{j=1}^n b_{p_j}^{*2} < 1$, andconversely, if $b$ is an $\ell_1$-multiplier for $\mathcal{H}_\infty$, then$|b_{p_j}| < 1$ 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 decreasingrearrangement of $b$).  Following an ingenious idea of Harald Bohr it turns out that this problem isintimately related with the question of characterizing those sequences $z$ inthe infinite dimensional polydisk $\mathbb{D}^\infty$ (the open unit ball of$\ell_\infty$) for which every bounded and holomorphic function $f$ on$\mathbb{D}^\infty$ has an absolutely convergent monomial series expansion$\sum_{\alpha} \frac{\partial_\alpha f(0)}{\alpha!} z^\alpha$. Moreover, westudy analogous problems in Hardy spaces of Dirichlet series and Hardy spacesof functions on the infinite dimensional polytorus $\mathbb{T}^\infty$.

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