Journal of the Ramanujan Mathematical Society
Volume 35, Issue 4, December 2020 pp. 341–345.
The maximal size of the k-fold divisor function for very large k
Authors:
Paul Pollack
Author institution:Department of Mathematics, University of Georgia, Athens, GA 30602
Summary:
Let d{k}(n) denote the number of ways of writing n as an
(ordered) product of k positive integers. When k=2, Wigert
proved in 1907 that
log d{k}(n) ≤ (1+o(1)) log k {log n}/{log log n}
(n → ∞ \infty). {*}
In 1992, Norton showed that (*) holds whenever k = o(log{n});
this is sharp, since (*) holds with equality when n is a product
of the first several primes. In this note, we determine the
maximal size of log d{k}(n) when k >> log {n}. To~illustrate:
Let κ > 0 be fixed, and let k,n → ∞ in such a way
that k/log{n} → κ; then
log d{k}(n) ≤ (s+κ ∑ {p prime} ∑ {l
≥ 1} {1}/{l p{l s}} + o(1)) log{n},
where s > 1 is implicitly defined by ∑ {p prime}
{log {p}}/{p{s}-1} = {1}/{κ}. Moreover, this upper
bound is optimal for every value of κ. Our results correct
and improve on recent work of Fedorov.
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