The last problem of Harald Bohr. (English) Zbl 0734.30003

Kahane, Jean-Pierre

J. Aust. Math. Soc., Ser. A 47, No. 1, 133-152 (1989).

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.

Reviewer: H.G.Feichtinger (Wien)

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

Keywords:

Harald Bohr; Dirichlet series; Lindelöf order function; almost sure convergence