an:01006728 Zbl 0881.11068 Queff??lec, H. H. Bohr's vision of ordinary Dirichlet series; old and new results EN J. Anal. 3, 43-60 (1995). 00036755 1995
j
11M41 40A05 11K99 43A46 60G15 60E15 30B50 Dirichlet series; probabilistic estimate; abscissa of uniform convergence; open questions 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 \textit{E. Hille} and \textit{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. S.L.Segal (Rochester) Zbl 0001.12901
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