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Mean values over certain sets of divisors of an integer. (Moyennes sur certains ensembles de diviseurs d’un entier.) (French) Zbl 0855.11050

For a given arithmetic function \(f(n)\) the authors consider the functions \[ \overline {f} (n)= {1\over {\tau (n)}} \sum_{d\mid n} f(d), \qquad \widehat {f} (n)= {1\over {2^{\omega (n)}}} \sum_{d \mid n} \mu^2 (d) f(d), \qquad \widetilde {f} (n)= {1\over {2^{\omega (n)}}} \sum _{\substack{ d\mid n\\ (d, n/d) =1}} f(d). \] Here \(\tau (n)\) is the number of positive divisors of \(n\), \(\mu (n)\) is the Möbius function, and \(\omega (n)\) is the number of distinct prime divisiors of \(n\). These functions represent the average of \(f\) over all of its divisors, squarefree divisors and unitary divisors, respectively. The authors study extensively the arithmetic properties of the new functions \(\overline {f}\), \(\widehat {f}\), \(\widetilde {f}\), and prove a number of interesting results. For example they show that, if \(f\) is additive (resp. multiplicative), then each of the functions \(\overline {f}\), \(\widehat {f}\), \(\widetilde {f}\) is also additive (resp. multiplicative). Mean values of the functions \(\overline {f}\), \(\widehat {f}\), \(\widetilde {f}\) are also studied, and it is is shown that, if \(f\) is strongly additive \((f (p^\alpha)= f(p)\) for all primes \(p\)), then \(f\) has a mean value if and only if \(\widetilde {f}\) or \(\widehat {f}\) has a mean value. The “discrepancy” function \[ \overline {\Delta} f(n):= {1\over {\tau (n)}} \sum_{d\mid n} (f(d)- \overline {f} (d)) \] (and analogously defined functions \(\widehat {\Delta} f(n)\), \(\widetilde {\Delta} f(n)\)) are introduced and studied. If \(f\) is additive, then each of the discrepancy functions is again additive. Finally some further generalizations and examples are also given.
Reviewer: A.Ivić (Beograd)

MSC:

11N37 Asymptotic results on arithmetic functions
11A25 Arithmetic functions; related numbers; inversion formulas
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