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.

The solution of Bohr’s problem (see Chapter 4) implies that for every Dirichlet series in \mathcal{H}_\infty, the sum ∑ |a_n| n^(-s) is finite for every Re s > 1/2, and we ask if we can in fact get to Re s=1/2. This is addressed by considering, for Dirichlet polynomials, the quotient between ∑ | a_n | and the norm (in \mathcal{H}_\infty) of the polynomial. We define S(x) as the supremum over all Dirichlet polynomials of length x ≥ 2 of these quotients. It is shown that S(x)=exp(- (1/\sqrt{2} + o(1)) (log n loglog n)^(1/2)) as x goes to ∞. This is reformulated in terms of the Sidon constant of the monomials as characters of the infinite-dimensional polydisc. The proof uses the hypercontractive Bohnenblust-Hille inequality and a fine decomposition of the natural numbers as those having ‘big’ and ‘small’ prime factors. Also, a version for homegeneous Dirichlet series is given.

Each Hardy space of Dirichlet series \mathcal{H}_p has an associated abscissa, and the analogue to Bohr’s problem arises in a natural way: to determine the maximal distance S_p between this abscissa and the abscissa of absolute convergence. If a Dirichlet series with coefficients (a_n) belongs to \mathcal{H}_p, then the series with coefficients (a_n/n^{ε}) belongs to \mathcal{H}_q for all q>p and ε >0. It is shown that S_p=1/2, and that, if we only consider m-homogeneous Dirichlet series, S_p^m=1/2. For every 1 ≤ p < ∞ the set of monomial convergence of the Hardy space H_p of functions on the infinite dimensional polytorus (hence also of the Hardy space H_2 on the infinite-dimensional polytorus) is l_2 ∩ Bc0. The space of all multipliers on the Hardy space of Dirichlet series \mathcal{H}_p coincides with \mathcal{H}_\infty.

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.

We give an alternative, probabilistic, approach to two of the subjects considered so far: the optimality of the exponent in the polynomial Bohnenblust-Hille inequality (see Chapter 6) and the lower bound for S in Bohr’s problem (see Chapters 1 and 4). We use a probabilistic device: the Kahane-Salem-Zygmund inequality. This shows that, for a given finite family of coefficients, a choice of signs can be found in such a way that the polynomial whose coefficients are the original ones multiplied by the signs has small norm (supremum on the polydisc). The proof uses Bernstein’s inequality and Rademacher random variables. We also look at the relationship between Rademacher and Steinhaus random variables, and deduce the classical Khinchin inequality from the Khinchin-Steinhaus inequality (see Chapter 6). We consider Dirichlet series, place signs before the coefficients, and define the almost sure abscissas (in each of the senses from Chapter 1) by considering each convergence for almost every choice of signs. An analogue of Bohr’s problem in this sense is considered.