Fan 2025 maximal order shifted prime divisor function

../

entropy_divisor_family: Uses Bernoulli-selected prime factors to construct exponentially many divisors in a narrow logarithmic window.

grh_divisor_family: Constructs the golden-ratio-sized family of comparable moduli and verifies the prime-progression estimate under GRH.

h_n_corollary: Uses Fermat's theorem to transfer the shifted-prime divisor lower bound to the coprimality threshold in Problem 820.

numerical_optimization: Proves uniqueness of the unconditional optimizer and certifies the quoted decimal constant with exact rational logarithm bounds.

proposition_3_1: Records the zero-free and smooth-modulus hypotheses that give a uniform lower bound for primes congruent to one.

representation_counting: Converts many comparable good divisors of a primorial into one integer with many distinct shifted-prime divisors.

theorem_1_1: Proves the unconditional 0.6736 log 2 lower bound and the golden-ratio lower bound under GRH.

unconditional_good_moduli: Removes exceptional conductors from the random divisor family and verifies every hypothesis of Harman's prime-progression theorem.

---

Kai (Steve) Fan and Paul Pollack, The maximal order of the shifted-prime divisor function, arXiv:2510.14167v1, 15 October 2025, twelve pages. Published in Integers 26A (2026), #A9, fourteen pages, doi:10.5281/zenodo.23017984.

Source version

The copy read for this card is the twelve-page arXiv v1 PDF, its canonical edition. The arXiv record was checked: it still listed only v1, named Steve Fan and Paul Pollack as authors, and supplied no journal reference. The same-day checks of Fan's publication page and Pollack's research page report that the paper has been accepted and is to appear in Integers, in honor of the eightieth birthdays of Melvyn Nathanson and Carl Pomerance. The journal site did not yet list a final article on that date. The journal published the paper on 28 September 2026 as Integers 26A (2026), #A9: the published file's first page prints receipt on 15 October 2025 and acceptance on 17 March 2026, and the volume's contents page lists the article (https://math.colgate.edu/~integers/vol26a.html).

The fourteen-page accepted author version, also read, has an Integers layout that says “#A1 INTEGERS 26 (2026),” but its received, revised, accepted, and published fields are blank. It is therefore evidence for the accepted form, not a final journal issue. A page-by-page comparison found the same theorem and proof argument as arXiv v1, with changed pagination and numbering, a new footnote crediting Gabdullin for an optimality observation about the auxiliary divisor family, acknowledgments, and a Granville reference. A comparison of text layers found the published file identical to it apart from the header, the affiliations, the filled dates, the placement of the dedication and the DOI line, so the two share pagination and numbering. Against v1, both renumber Theorem 1.1 as Theorem 1, Proposition 3.1 as Proposition 1, and equations (1.1)--(1.5), (2.1)--(2.2), (3.1)--(3.18) and (4.1)--(4.4) as (1)--(5), (6)--(7), (8)--(25) and (26)--(29). Both retain the two proof-text slips described below, so neither supplies a mathematical correction requiring a change of canonical version. Result pages cite the versioned arXiv v1 pagination. The arXiv record (https://arxiv.org/abs/2510.14167v1, read 2026-10-07) names the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 license for that version. The accepted author version prints no copyright or license line on any of its fourteen pages. Its bytes match the PDF that the paper's entry on the second author's research page links, http://www.pollack-math.net/maxomegastar.pdf, and that page states no terms (https://www.pollack-math.net/research.html, read 2026-10-07); the first author's publication page links only the arXiv version and states nothing beyond its footer "© 2026 Steve Fan" (https://stvfan.github.io/publications/, read 2026-10-02), and the journal's statement "All works of this journal are licensed under a Creative Commons Attribution 4.0 International License" (https://math.colgate.edu/~integers/, read 2026-10-02) covers its published file, not this accepted form; the term is unstated. The Zenodo record of the published file names the Creative Commons Attribution 4.0 International license (https://zenodo.org/records/23017984, read 2026-10-07).

This directory is the single canonical home for the source. The proposed connection to Problem 345 came from a forum survey rather than this paper and is not recorded as a result of the source.

Main result and proof route

For

$$\omega^*(n)=\#\{p\text{ prime}:p-1\mid n\},$$

Theorem 1.1 proves that infinitely many $n$ satisfy

$$\omega^*(n)> \exp\!\left(0.6736\log 2\,\frac{\log n}{\log\log n}\right)$$

without an unproved hypothesis. Under the Generalized Riemann Hypothesis for Dirichlet $L$-functions, infinitely many $n$ satisfy

$$\omega^*(n)> \exp\!\left(\left(\log\frac{1+\sqrt5}{2}+o(1)\right) \frac{\log n}{\log\log n}\right).$$

The complete same-paper route is split into the representation-counting lemma, the Bernoulli divisor-family lemma, the GRH family, and the unconditional good-modulus construction. The latter invokes the exact external Harman proposition. The optimization page proves uniqueness of the optimizing parameter and gives an exact-rational interval certificate for the decimal $0.6736$.

Consequence for $H(n)$

The [[integer_sequences/fan_2025_maximal_order_shifted_prime_divisor_function/h_n_corollary|derived $H(n)$ corollary]] combines Theorem 1.1 with the canonical Fermat/product bound. For integers $n\ge2$, let $H(n)$ be the least $l\ge3$ for which some integer $k$ with $2\le k<l$ has $\gcd(k^n-1,l^n-1)=1$; the canonical threshold page proves that this minimum exists. Then infinitely many $n$ obey

$$H(n)>\exp\!\left(n^{0.6736\log 2/\log\log n}\right).$$

This elementary implication is recorded for Problem 820. It is not a labeled Fan--Pollack result: the underlying connection belongs to the Problem 820 history in Erdős's 1974 discussion and was pointed out for this new bound in van Doorn's July 2026 site comment.

External-input and coverage boundary

The GRH prime-number theorem in progressions, Harman's 2008 theorem, the exceptional-zero statement and Montgomery zero-density estimate used to discard bad conductors, and standard prime- and divisor-function estimates are stated at the exact strength used. Their original proofs are external and are not recursively reconstructed. Conditional conclusions remain explicitly conditional.

Sections 2 and 4 of the paper also discuss Prachar's older construction, a Density-Hypothesis extension, and consequences of Pomerance's smooth shifted-prime conjecture. Those contextual results are not needed for Theorem 1.1 and are not claimed as additional full-proof records here. In the unconditional proof, equation (3.17) of v1 (equation (24) of the accepted version) claims an $L^{2/3}$ logarithmic window, whereas the earlier displayed prime-number-theorem error directly supplies the wider $O(L/(\log L)^2)$ window. The good-modulus page uses that justified width; it is already more than sufficient for (3.14), Harman's ranges, and the entropy count. In the last argument, v1 p. 10 (accepted-version p. 12) prints $n=(m-1)p$ where the counted pairs and the following page use $n=m(p-1)$; the latter is the expression that is $y$-smooth. Neither printed issue changes the compiled theorem or the $H(n)$ corollary.

All twelve arXiv-v1 pages and all fourteen accepted-author pages were rendered and visually inspected. The result pages provide complete source-based reconstructions at the explicit external-input boundaries stated above.

Bears on. Problem 820 (through the [[integer_sequences/fan_2025_maximal_order_shifted_prime_divisor_function/h_n_corollary|derived $H(n)$ corollary]], the unconditional bound of Theorem 1.1 gives $H(n)>\exp(n^{0.6736\log2/\log\log n})$ for infinitely many $n$, a lower bound of the form the problem's estimate asks for, with the constant $0.6736\log2$; it gives no upper bound for $H(n)$ and does not decide whether $H(n)=3$ infinitely often).

No file of this source is held. Neither the arXiv edition nor the accepted author version is under an open license; the published file's CC BY 4.0 license is open, but that file is not held. The card cites the edition it names above.