Completely multiplicative functions of sum zero. (Fonctions complètement multiplicatives de somme nulle.) (French. English summary) Zbl 1405.11050
Summary: Let \(f\) be a function from \(\mathbb{N}^\ast\) to \(\mathbb{C}\) not identically 0. We call it \(C M O\) if \(f(a b) = f(a) f(b)\) for all \(a\) and \(b\) and \(\sum_{n = 1}^{+ \infty} f(n) = 0\). We give properties and examples of \(C M O\) functions. A basic example goes back to Euler, namely \(f(p) = - 1/p\) for every prime number \(p\). We study how far from this example the \(C M O\) character is kept. The zeroes of Dirichlet series are implied in this study, as well as in other examples. The relation between \(C M O\) and the generalized Riemann hypothesis is pointed out at the end of the article.
MSC:
| 11F32 | Modular correspondences, etc. |
| 11M45 | Tauberian theorems |
| 11N80 | Generalized primes and integers |
References:
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