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Deux résultats de théorie probabiliste des nombres. (Two results in probabilistic number theory). (French) Zbl 0766.11036

An arithmetic semigroup is a normed semigroup such that \(\sum_{a\in A, N(a)\leq x} 1=LX+o(x)\) with some constant \(L>0\). The first result asserts that the Turán-Kubilius inequality may fail in such a semigroup, even in the following strong sense: every arithmetic semigroup has an extension in which the T-K inequality fails. The second result is a generalization of a theorem of Elliott on the moments of Ramanujan’s \(\tau\) function. The results are announced without proof.

MSC:

11K99 Probabilistic theory: distribution modulo \(1\); metric theory of algorithms
11K65 Arithmetic functions in probabilistic number theory
11N45 Asymptotic results on counting functions for algebraic and topological structures
11N80 Generalized primes and integers
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