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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.

MSC:
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
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