zbMATH — the first resource for mathematics

H. Bohr’s vision of ordinary Dirichlet series; old and new results. (English) Zbl 0881.11068
The author discusses briefly a theorem of Harald Bohr from 1913: \(\sum_{n=1}^\infty a_nn^{-s}\) has a finite abscissa of uniform convergence \(\sigma_u\) implies \(\sum_{p\text{ prime}}|a_p|p^{-s}\) converges for \(\sigma>\sigma_u\), and a theorem of Hewitt and Williamson about ordinary Dirichlet series representations of the reciprocals of absolutely convergent Dirichlet series. He proves a substantial generalization of Bohr’s result which leads to a probabilistic estimate connected with the determination of the abscissa of uniform convergence.
The author also obtains a new proof of the result of E. Hille and H. F. Bohnenblust [Ann. Math., II. Ser. 32, 600-622 (1931; Zbl 0001.12901)] that Bohr’s estimate \(\sigma_a- \sigma_u\leq 1/2\) (where \(\sigma_a\) is the abscissa of absolute convergence and \(\sigma_u\) as above) is best possible. This is based on a new result improving an earlier one of his. He also gives a novel proof of the result of Hewitt and Williamson using Bohr’s “vision”. The paper closes with some open questions.

11M41 Other Dirichlet series and zeta functions
40A05 Convergence and divergence of series and sequences
11K99 Probabilistic theory: distribution modulo \(1\); metric theory of algorithms
43A46 Special sets (thin sets, Kronecker sets, Helson sets, Ditkin sets, Sidon sets, etc.)
60G15 Gaussian processes
60E15 Inequalities; stochastic orderings
30B50 Dirichlet series, exponential series and other series in one complex variable