J Austral Math Soc Ser A 47 pp133--152, 1989.

The Last Problem of Harold Bohr

Jean-Pierre Kahane

Abstract

The first and last papers of Harold Bohr deal with ordinary Dirichlet series å₁^¥ a_nn^-3 and their order (or Lindelöf) function m(s) (= inf{a; f(s+ it) = o(|t|^a)}). The Lindelöf hypothesis is m(s) = inf(0, ½ - t) when a_n = (-1)ⁿ. Are there ordinary Dirichlet series with -1 < m¢(s) < 0 for some s? A negative answer would imply Lindelöf's hypothesis. This is the last problem of Harold Bohr. This paper gives (1) a review on Bohr's results on ordinary Dirichlet series; (2) a series å₁^¥ ±n^-1 with the solution of a previous problem of Bohr, (3) the following answer to the last problem: m¢(s) can approach -½, and necessarily m(s+ m(s) + ½) = 0. The characterization of the order functions of ordinary Dirichlet series remains an open question.

1980 AMS Subject Classification (1985 Revision): 30B50

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Authors

Jean-Pierre Kahane

Département de Mathématique, Université de Paris-Sud, 91405 Orsay, France.