LAUSR

Let $${\mathscr {H}}_\infty $$H∞ be the set of all ordinary Dirichlet series $$D=\sum _n a_n n^{-s}$$D=∑nann-s representing bounded holomorphic functions on the right half plane. A completely multiplicative sequence $$(b_n)$$(bn) of complex numbers is said to be an $$\ell _1$$ℓ1-multiplier for $${\mathscr {H}}_\infty $$H∞ whenever $$\sum _n |a_n b_n| < \infty $$∑n|anbn|<∞ for every $$D \in {\mathscr {H}}_\infty $$D∈H∞. We study the problem of describing such sequences $$(b_n)$$(bn) in terms of the asymptotic decay of the subsequence $$(b_{p_j})$$(bpj), where $$p_j$$pj denotes the jth prime number. Given a completely multiplicative sequence $$b=(b_n)$$b=(bn) we prove (among other results): b is an $$\ell _1$$ℓ1-multiplier for $${\mathscr {H}}_\infty $$H∞ provided $$|b_{p_j}| < 1$$|bpj|<1 for all j and $$\overline{\lim }_n \frac{1}{\log n} \sum _{j=1}^n b_{p_j}^{*2} < 1$$lim¯n1logn∑j=1nbpj∗2<1, and conversely, if b is an $$\ell _1$$ℓ1-multiplier for $${\mathscr {H}}_\infty $$H∞, then $$|b_{p_j}| < 1$$|bpj|<1 for all j and $$\overline{\lim }_n \frac{1}{\log n} \sum _{j=1}^n b_{p_j}^{*2} \le 1$$lim¯n1logn∑j=1nbpj∗2≤1 (here $$b^*$$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 $${\mathbb {D}}^\infty $$D∞ (the open unit ball of $$\ell _\infty $$ℓ∞) for which every bounded and holomorphic function f on $${\mathbb {D}}^\infty $$D∞ has an absolutely convergent monomial series expansion $$\sum _{\alpha } \frac{\partial _\alpha f(0)}{\alpha !} z^\alpha $$∑α∂αf(0)α!zα. Moreover, we study analogous problems in Hardy spaces of Dirichlet series and Hardy spaces of functions on the infinite dimensional polytorus $${\mathbb {T}}^\infty $$T∞.