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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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