The text is closed by coming back to Bohr’s absolute convergence problem, this time for vector-valued Dirichlet series. For a Banach space X abscissas and strips S(X) and S_p(X), analogous to those defined in Chapters 1 and 12 are considered. It is shown that all these strips equal 1-1/cot(X), where cot(X) is the optimal cotype of X.

We give the solution of Bohr’s problem, showing that in fact S=1/2. This is done by considering an analogous problem where only m-homogeneous Dirichlet series are taken into account (defining, then, S^m). Using the isometry between homogeneous Dirichlet series and polynomials, the problem is translated into a problem for these. For each m we produce an m-homogeneous polynomial P such that for every q > (2m)/(m-1) there is a point z in l_q for which the monomial series expansion of P does not converge at z. This shows that, contrary to what happens for finitely many variables, holomorphic functions in infinitely many variables may not be analytic. This also shows that (2m)/(m-1) ≤ S^m for every m and then gives the result. There is more. For each fixed 0 ≤ σ ≤ 1/2 there is a Dirichlet series whose abscissas of uniform and absolute convergence are at distance exactly σ.

We establish the basic notions around Dirichlet series that are going to be used all along the text. A Dirichlet series converges on half-planes, and that there it defines a holomorphic function. For a given Dirichlet series we consider four abscissas definining the maximal half-planes on which it: converges, defines a bounded holomorphic function, converges uniformly or converges absolutely. We formulate the problem of determining the maximal possible distance between these abscissas. The difference between the abscissa of convergence and absolute convergence is at most one, and this is attained. Also, the abscissa of uniform convergence and of boundedness always coincide (this is Bohr theorem). Then Bohr’s problem is established: to determine S, the maximal possible width of the strip of absolute but not uniform convergence of Dirichlet series, and we show that it is at most 1/2. Finally we introduce the Banach space \mathcal{H}_\infty of Dirichlet series that converge and define a bounded holomorphic function on the right half-plane and reformulate Bohr’s problem in terms of this space. This becomes later an important tool.