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The last problem of Harald Bohr

Part of: Series expansions

Published online by Cambridge University Press:  09 April 2009

Jean-Pierre Kahane
Jean-Pierre Kahane
Affiliation:
Départment de Mathématique, University of Paris-Sud, 91405 Orsay, France
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Abstract

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The first and last papers of Harald Bohr deal with ordinary Dirichlet series and their order (or Lindelöf) function μ(σ) (= inf{α;f(σ + it) + 0(|t|α)}). The Lindelöf hypothesis is μ(σ) = inf(0, ½ − t) when an = (−1)n. Are there ordinary Dirichlet series with −l < μ′ (σ) < 0 for some σ? A negative answer would imply Lindelöf's hypothesis. This is the last problem of Harald Bohr. This paper gives (1) a review on Bohr's results on ordinary Dinchlet series; (2) a review on results of the author and of Queffelec on “almost sure” and “quasi sure” properties of series with the solution of a previous problem of Bohr; (3) the following answer to the last problem: μ′(σ) can approach − ½, and necessarily μ(σ + μ(σ) + ½) = 0. The characterization of the order functions of ordinary Dirichlet series remains an open question.

MSC classification

Secondary: 30B50: Dirichlet series and other series expansions, exponential series
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 47 , Issue 1 , August 1989 , pp. 133 - 152
Copyright
Copyright © Australian Mathematical Society 1989

References

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