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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 . 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 function in the usual sense carried by a lacunary set of integers (theorem 1).
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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