Multiplicative partitions of numbers with a large squarefree divisor
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Multiplicative partitions of numbers with a large squarefree divisor Paul Pollack1 Received: 31 December 2018 / Accepted: 18 June 2019 © Springer Science+Business Media, LLC, part of Springer Nature 2019
Abstract For each positive integer n, let f (n) denote the number of multiplicative partitions of n, meaning the number of ways of writing n as a product of integers larger than 1, where the order of the factors is not taken into account. It was shown by Oppenheim (J Lond Math Soc 1:205–211, 1926) that, as x → ∞, max
n≤x n squarefree
where L(x) = exp log x ·
f (n) = x/L(x)2+o(1) ,
log log log x log log x .
Without the restriction to squarefree n, the
maximum is the significantly larger quantity x/L(x)1+o(1) ; this was proved by Canfield et al. (J Number Theory 17:1–28, 1983). We prove the following theorem that interpolates between these two results: for each fixed α ∈ [0, 1], max
n≤x rad(n)≥n α
f (n) = x/L(x)1+α+o(1) .
We deduce, on the abc conjecture, a nontrivial upper bound on how often values of certain polynomials appear in the range of Euler’s ϕ-function. Keywords Multiplicative partition · Factorization · Divisor functions · Factorisatio numerorum · abc conjecture Mathematics Subject Classification Primary 11A51 · Secondary 11N32 · 11N56 · 11N64
The author is supported by NSF Award DMS-1402268.
B 1
Paul Pollack [email protected] Department of Mathematics, University of Georgia, Athens, GA 30602, USA
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P. Pollack
1 Introduction By a multiplicative partition (or unordered factorization) of n, we mean a way of decomposing n as a product of integers larger than 1, where two decompositions are considered the same if they differ only in the order of the factors. Let f (n) denote the number of multiplicative partitions of n. For example, f (12) = 4, corresponding to the factorizations 2 · 6, 2 · 2 · 3, 3 · 4, and 12. The function f (n) was introduced by MacMahon in 1923 and was shortly afterwards the subject of two papers by Oppenheim [11,12]. The main result of Oppenheim’s first paper concerns the maximum size of f (n). Let logk x denoting the kth iterate of the natural logarithm, and put log3 x . L(x) = exp log x · log2 x In [11], Oppenheim claims to prove that f (n) ≤ n/L(n)2+o(1) , as n → ∞, and that this is optimal: there is an infinite, increasing sequence of positive integers n along which f (n) = n/L(n)2+o(1) . However, in 1983, Canfield et al. [2] disproved Oppenhein’s “theorem”, showing that the true maximal order is n/L(n)1+o(1) ; more precisely, max f (n) = x/L(x)1+o(1) ,
(1)
n≤x
as x → ∞. Oppenheim’s “proof” that f (n) ≤ n/L(n)2+o(1) rests on a mistaken assertion concerning the maximal order of the k-fold divisor function dk (n). Specifically, Oppenheim claims that dk (n) < k log n/ log2 n+log n/(log2 n)
2 +O(log n/(log n)3 ) 2
,
(2)
for all large n and all k. Now (2) is true when k = 2 (a result of Ramanujan [17]), and in fact true for each fixed k (see [4] for sharper results), but it is not true uniformly in k, and this invalidates Oppenheim’s argument. Upper bounds
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