JournalslemVol. 63, No. 1/2pp. 155–164

Sur l’exemple d’Euler d’une fonction complètement multiplicative de somme nulle

  • Jean-Pierre Kahane

    Université Paris-Sud, Orsay, France
  • Eric Saias

    Université Pierre et Marie Curie, Paris, France
Sur l’exemple d’Euler d’une fonction complètement multiplicative de somme nulle cover

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Abstract

In 1737 Euler introduced a series whose general term is the first example of a completely multiplicative function whose sum is 0, what we write CMOCMO. Euler proved that the sum of his series is 0, assuming that the sum exists. The convergence of the series was proved later, as a companion of the prime number theorem. We consider the same problem for generalized primes and integers in the sense of Beurling 1937. A key is a theorem of Diamond 1977, which gives a condition on the generalized primes in order that the generalized integers have a density. According to Diamond’s condition the analogue of the Euler series converges and its sum is 0 (theorem 2). That is a way (and the only way as far as we can guess) to construct a CMOCMO function in the usual sense carried by a lacunary set of integers (theorem 1).

Cite this article

Jean-Pierre Kahane, Eric Saias, Sur l’exemple d’Euler d’une fonction complètement multiplicative de somme nulle. Enseign. Math. 63 (2018), no. 1, pp. 155–164

DOI 10.4171/LEM/63-1/2-4