which improves an earlier result of Dornhoff and Spitznagel for S(x). The main tool is de Bruijn's well-known result on the number of integers not exceeding x, all whose prime factors are not greater than y. It is conjectured that
Part II discusses a problem due to Hadwiger: Let D(n) denote the set of integers with the property that if k in D(n) then the n dimensional unit cube can be decomposed into k homothetic n dimensional cubes. Let c(n) be the smallest integer such that k \geq c(n) implies k in D(n). It is proved that
and a number of related number theoretical problems are discussed.
Part III is devoted to the functions \sigma and \phi. Several previous results of the author are mentioned and, as goes without saying for a paper of Erdös's, various unsolved problems are stated. Finally, this part of the paper contains some new results with hints of their proofs. We mention one:
with \nu the function that counts the different prime factors of n.
Classif.: * 11N37 Asymptotic results on arithmetic functions
11A25 Arithmetic functions, etc.
11-02 Research monographs (number theory)
20D05 Classification of simple and nonsolvable finite groups
11H99 Geometry of numbers
00A07 Problem books
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