On Euler's example of a completely multiplicative function with sum 0 (Q5916044)

Summary: 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 \(CMO\). 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 \textit{A. Beurling} [Acta Math. 68, 255--291 (1937; Zbl 0017.29604; JFM 63.0138.01) ]. A key is a theorem of \textit{H. G. Diamond} [J. Reine Angew. Math. 295, 22--39 (1977; Zbl 0355.10038)], 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 \(CMO\) function in the usual sense carried by a lacunary set of integers (theorem 1).