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The last problem of Harald Bohr. (English) Zbl 0734.30003
The author takes the centenary of Harald Bohr (1887-1951) as opportunity to review some of Bohr’s work (early and late) on Dirichlet series of the form $$f(s)=\sum^{\infty}_{n=1}a_ nn^{-s}$$, and their Lindelöf order function $\mu (s)=\inf \{\alpha | \quad f(s+it)\}=o(| t|)^{\alpha}).$ The general theme are convergence and summability properties of the series and properties of f as analytic functions, with special emphasis on series arising from sequences $$a_ n=\pm 1$$ and the question whether $$\mu (1/2)=0$$ for the case of an alternating sequence. Partial answers are shown to follow from earlier results by J. P. Kahane [C. R. Acad. Sci., Paris, Sér. A 276, 739-742 (1973; Zbl 0252.30007)] and H. Queffélec [Can. J. Math. 32, 531-558 (1980; Zbl 0475.30006)] on almost sure convergence of such Dirichlet series.

##### MSC:
 30B50 Dirichlet series, exponential series and other series in one complex variable 42A75 Classical almost periodic functions, mean periodic functions 42A55 Lacunary series of trigonometric and other functions; Riesz products