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Arithmetical semigroups. V: Multiplicative functions. (English) Zbl 0809.11059

The author continues his deep studies of an arithmetical semigroup \((G,\partial)\) satisfying Axiom \(\text{A}^ \#\), as defined in the reviewer’s monograph [“Analytic arithmetic of algebraic function fields”, M. Dekker (1979; Zbl 0411.10001)]. Here he develops counterparts of well known theorems of E. Wirsing and G. Halász on multiplicative functions of natural numbers for multiplicative functions \(f\) on \(G\). In particular it is shown that an asymptotic mean-value or closely related limit always exists for \(| f|\leq 1\) if some subsidiary conditions apply. Also, without assuming Axiom \(\text{A}^ \#\), if \[ \sum_{\text{prime }p,\;\partial(p)=m} f(p)= {{q^ m} \over m} (\tau+ o(1)) \quad \text{as } m\to\infty, \] for constants \(q>1\), \(\tau\geq 0\), and a subsidiary condition holds, then \[ \sum_{\partial(a)=n} f(a)= {{q^ n} \over n} \Biggl( {{e^{- \gamma\tau}} \over {\Gamma(\tau)}}+ o(1) \Biggr) \prod_{\partial(p) \leq n} \Biggl(1+ \sum_{r=1}^ \infty f(p^ r) q^{-\partial(p^ r)} \Biggr); \] here \(\gamma\) is Euler’s constant.

MSC:

11N80 Generalized primes and integers
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References:

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