On the frequencies of large values of divisor functions

../

theorem_1_11: Bounds integers with many divisors using an explicit one-sided exponential estimate.

---

Karl K. Norton, On the frequencies of large values of divisor functions, Acta Arithmetica 68 (1994), no. 3, 219–244, DOI 10.4064/aa-68-3-219-244. The canonical published PDF was acquired from the journal archive on 2026-09-05. The file's text layer carries no copyright or license line; the publisher's record (https://www.impan.pl/get/doi/10.4064/aa-68-3-219-244, read 2026-10-02) offers the PDF under the link "Pobierz zgodnie z CC-BY" ("Free download under CC-BY license" on the English site), a Creative Commons Attribution license whose version the record does not name; the site footer "Copyright © 2026 by IMPAN. All rights reserved." speaks for the site, not the article.

The ordinary-divisor specialization of Theorem 1.11 is recorded as an exact external input. Its full proof is not yet compiled. The source studies the generalized divisor function $d_z(n)$, with $d_2(n)=\tau(n)$; it distinguishes strict and non-strict upper tail counts.

Read status: claims checked for Theorem 1.11 (printed p. 221) and the definitions (1.1)--(1.4) and the $\log_k$ and $O$ conventions it uses (pp. 219--220), read clause by clause on the page images; the proof was not read.

The selected estimate is sufficient for McNew–Setty's primitive-covering count, without relying on a broad two-sided asymptotic across all thresholds. Other estimates and proofs in the paper remain to be compiled.

Bears on

• Problem 7: an analytic input for the number of primitive covering periods, not a resolution of odd-covering existence.