Publications of (and about) Paul Erdös
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 sum2 \leq n \leq x1/\beta(n) and sum2 \leq n \leq x1/B(n). Theorem 2 estimates sum2 \leq n \leq x\beta(n)/B(n) and sum2 \leq n \leq xB(n)/\beta(n) in the form of x+O(x \exp(-C(log x log log x) ½ )). Theorem 3 estimates sumn \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.
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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