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An Asymptotic Formula for Reciprocals of Logarithms of Certain Multiplicative Functions

Published online by Cambridge University Press:  20 November 2018

Jean-Marie de Koninck  and
Aleksandar Ivić
Jean-Marie de Koninck
Affiliation:
Départment de Mathématiques, Université Laval, Québec, P.Q. Canada, G IK 7P4
Aleksandar Ivić
Affiliation:
Rudarsko-Geološki Fakultet, Djušina 7, 11000 Beograd, Yugoslavia
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Sums of the form where f(n) is a multiplicative arithmetical function and denotes summation over those values of n for which f(n)>0 and f(n) ≠1, were studied by De Koninck [2], De Koninck and Galambos [3], Brinitzer [1] and Ivič [5]. The aim of this note is to give an asymptotic formula for a certain class of multiplicative, positive, primeindependent functions (an arithmetical function is prime-independent if f(pv) = g(v) for all primes p and v = 1, 2, …). This class of functions includes, among others, the functions a(n) and τ(e)(n), which represent the number of nonisomorphic abelian groups of order n and the number of exponential divisors of n respectively, and none of the estimates of the above-mentioned papers may be applied to this class of functions. We prove the following.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 21 , Issue 4 , 01 December 1978 , pp. 409 - 413
Copyright
Copyright © Canadian Mathematical Society 1978

References

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