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On Euler's example of a completely multiplicative function with sum 0

Abstract : Euler published a formula that now reads View the MathML source∑1∞λ(n)n=0, λ being the completely multiplicative function equal to −1 on the prime numbers. Thus View the MathML source(λ(n)n) is an example of a CMO function (completely multiplicative with sum 0). We extend this formula by considering λ as defined on Beurling's generalized prime numbers and integers, according to Diamond's condition on generalized primes, which implies a regular distribution of the generalized integers (théorème 3). As an application, we show how to contruct a CMO function carried by a set of integers whose counting function is of the form Dxa(1+o(1))Dxa(1+o(1))(x→∞)(x→∞), for any given a between 0 and 1 (théorème 1).
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Jean-Pierre Kahane, Eric Saias. On Euler's example of a completely multiplicative function with sum 0. Comptes Rendus Mathématique, Elsevier Masson, 2016, ⟨10.1016/j.crma.2016.03.009⟩. ⟨hal-01304311⟩



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