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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.

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