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.

A holomorphic function f on the disc has a Taylor expansion with coefficients c_k. Bohr asked about the maximal 0<r<1 so that the supremum for |z|<r of ∑ | c_k z^k | is less than or equal to the supremum for |z|<1 of |f(z)|. Bohr’s power series theorem answers this question showing that r=1/3 is best possible. The n-th Bohr radius K_n is defined as the best r for which an analogous question holds for holomorphic functions on the n-dimensional polydisc. The sequence (K_n) is decreasing and tends to 0 as n goes to ∞ asymptotically like (\log n/n)^(1/2). The proof os this relies on an improved version of the polynomial Bohnenblust-Hille inequality (see Chapter 6), where the constant grows at most exponentially, and to get this a Khinchin-Steinhaus inequality for polynomials is needed, showing that all L_p norms of polynomials in n variables are equivalent.