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**Zentralblatt MATH**

**Publications of (and about) Paul Erdös**

**Zbl.No: ** 463.10032

**Autor: ** De Koninck, J.-M.; Erdös, Paul; Ivic, A.

**Title: ** Reciprocals of certain large additive functions. (In English)

**Source: ** Can. Math. Bull. 24, 225-231 (1981).

**Review: ** Let \beta(n) be the sum of distinct prime divisor of n and B(n) the sum of all prime factors of n (counting multiples). Methods are used that can be as well applied to several pairs of similarly related large additive functions, to prove three theorems. Theorem 1 gives upper and lower bounds on **sum**_{2 \leq n \leq x}1/\beta(n) and **sum**_{2 \leq n \leq x}1/B(n). Theorem 2 estimates **sum**_{2 \leq n \leq x}\beta(n)/B(n) and **sum**_{2 \leq n \leq x}B(n)/\beta(n) in the form of x+O(x \exp(-C(log x log log x)^{ ½ })). Theorem 3 estimates **sum**_{n \leq x}'1/(B(n)-\beta(n)) as Cx+O(x^{ ½ } log x). The constant C in theorem 3 is given explicitly and **sum**' denotes the sum over those n for which B(n)\neq\beta(n). Most of the method is elementary but an analytical method using the Riemann zeta-function is involved in the proof of theorem 3.

**Reviewer: ** J.P.Tull

**Classif.: ** * 11N37 Asymptotic results on arithmetic functions

11N05 Distribution of primes

**Keywords: ** reciprocals of large additive functions; sum of distinct prime divisors; sum of all prime divisors; partitions; asymptotic estimates

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