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Moments of the shifted prime divisor function

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Pith's one-line read For each $k\ge 2$, the $k$-th moment of the shifted prime divisor function is comparable to $x(\log x)^{2^k-k-1}$, confirming the Fan–Pomerance conjecture.

desk verdict The lower bound and the lcm identity are solid, but a subset-counting error in Lemma 4.1 leaves the upper bound one log factor short, so the main theorem is not yet proved. read the letter →

arxiv 2505.24050 v1 pith:KCUYXYRQ submitted 2025-05-29 math.NT

Mikhail R. Gabdullin This is my paper
classification math.NT MSC 11A4111N3611N37
keywords shiftedprimesdivisorfunctionleastcommonmultiplemomentsmultiplicativefunctionsBrunsieveGallagherprimenumbertheoremFan-Pomeranceconjecture
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper proves that for every fixed integer $k\ge 2$, the sum over $n\le x$ of $\omega^*(n)^k$, where $\omega^*(n)$ counts divisors of $n$ of the form $p-1$ with $p$ prime, is comparable to $x(\log x)^{2^k-k-1}$. This is exactly the order predicted by Fan and Pomerance for all $k$, previously known only for $k=2,3$. The result matters because it pins down the higher moments of a divisor-like function whose average matches $\omega(n)$ but whose variance and higher moments behave differently, and it supplies the first nontrivial upper bounds for $k>3$. The proof works with prime $k$-tuples whose shifted values have small least common multiple, using a multiplicative inclusion-exclusion identity for the lcm and sieve estimates.

What carries the argument

The engine of the proof is the lcm identity of Lemma 2.1: $\operatorname{lcm}(t_1,\ldots,t_k)$ equals the product over odd-size subsets of gcds of the corresponding $t_i$ divided by the product over even-size subsets. This turns the condition $\operatorname{lcm}(p_1-1,\ldots,p_k-1)\leq x$ into an inequality involving pairwise gcds $m_{ij}$, which can be controlled by multiplicative functions. For the upper bound, the paper uses the finer decomposition into pairwise coprime factors $u_S$ defined by $u_S = t_S \prod_{T\supsetneq S} u_T^{-1}$, so that every gcd of sub-products splits as a product of $u$'s; a Brun sieve then bounds the number of choices for the largest factor, and the auxiliary sum over the remaining factors is handled by Lemma 4.1.

What would settle it

Compute the sum in (4.12) numerically for $k=4$, $\nu=1$ on a fine grid of $u_S$ near $x$, and compare with $C(\log x)^5$; a violation for any fixed large $x$ would disprove the uniformity lemma. Alternatively, evaluate $\sum_{n\leq x}\omega^*(n)^4$ directly for $x$ up to, say, $10^9$ and check that the ratio to $x(\log x)^{11}$ stays bounded away from $0$ and infinity; a ratio tending to $0$ or infinity would contradict Theorem 1.1.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is Theorem 1.1: for each integer $k\ge 2$, $\sum_{n\leq x}\omega^*(n)^k \asymp x(\log x)^{2^k-k-1}$, with implied constants depending only on $k$. The proof reduces the moment sum to counting $k$-tuples of primes $(p_1,\ldots,p_k)$ with $\operatorname{lcm}(p_1-1,\ldots,p_k-1)\leq x$, and establishes the matching lower and upper bounds for that count, $P_k(x)\asymp x(\log x)^{2^k-k-2}$. The lower bound comes from a new identity for the least common multiple, written as a ratio of products of gcds of subsets of odd and even cardinality, combined with Gallagher's prime number theorem and a Landau–Page argument that avoids exceptional characters. The upper bound generalizes the method Fan and Pomerance used for $k=3$: a hierarchical factorization of the lcm into pairwise-coprime factors $u_S$, a Brun sieve count for the largest factor, and a uniform estimate for an auxiliary sum over these factors.

Load-bearing premise

The upper bound depends on Lemma 4.1, which asserts that a certain sum over the auxiliary factors $u_S$ is $O((\log x)^{2^k-\nu-1})$ uniformly in all previously chosen data; the proof of that uniformity invokes a constant $L$ that is merely required to be large depending on $k$, and if that uniformity fails the whole upper bound would not follow.

Editorial extensions

If this is right

  • For every fixed $k\ge 2$, the moment $\sum_{n\leq x}\omega^*(n)^k$ has the exact order $x(\log x)^{2^k-k-1}$, so the Fan–Pomerance conjecture is fully confirmed.
  • The proof yields the first nontrivial upper bounds for $k>3$, replacing the trivial bound $\omega^*(n)\leq \tau(n)$.
  • Markov's inequality applied to the new moment bounds gives distribution estimates $N(x,y)=\#\{n\leq x:\omega^*(n)\geq y\}$ of the form in Proposition 5.1.
  • The concentration arguments in Section 5 show that for ranges such as $y\asymp \log x/\log_2 x$ and $y\asymp (\log x)^{2j-1-1}$, the distribution estimates are sharp up to a $\log_2 x$ factor.
  • The lcm identity and the $u_S$ decomposition provide reusable tools for counting shifted-prime configurations beyond the moment problem.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • An editor's extension: the same decomposition might yield asymptotic constants $c_k$, since the proof already isolates the main multiplicative sums whose mean values are evaluated; determining $c_k$ would require tracking the constants throughout the sieve steps.
  • Another extension left implicit: the $u_S$ factorization and Lemma 4.1 could apply to neighbouring counting problems for shifted primes, such as tuples with prescribed gcd patterns, not just the moment sum.
  • A testable consequence: for $y$ in the intermediate range $(\log x)^{2j-1-1}\ll y \ll (\log x)^{2j-1}$, Proposition 5.2 predicts $N(x,y)$ has size $x(\log x)^{2j-j-1}/(y^j \log_2 x)$ infinitely often; a direct computation for $j=3$ and moderate $x$ would let one see whether the predicted thickness appears.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

1 major / 5 minor

Summary. The paper studies the shifted prime divisor function ω^*(n) = #{d|n : d = p−1, p prime}. It claims to prove that for every fixed integer k ≥ 2, ∑_{n≤x} ω^*(n)^k ≍ x(log x)^{2^k−k−1}, confirming a conjecture of Fan and Pomerance. The proof reduces the moment estimate to bounding P_k(x) = #{(p_1,…,p_k) : [p_1−1,…,p_k−1] ≤ x}. The lower bound uses a new combinatorial identity for the least common multiple, Gallagher’s prime number theorem with a Siegel-zero exclusion, and Wirsing’s mean-value theorem. The upper bound uses induction on k, a decomposition of the lcm into multiplicative pieces, Brun’s sieve, and a theorem of Pollack on multiplicative functions on sifted sets. The final section gives distribution estimates for N(x,y) = #{n ≤ x : ω^*(n) ≥ y}.

Significance. If correct, the result settles the Fan–Pomerance conjecture for every fixed k, going well beyond the previously known cases k = 2, 3. The combinatorial identity for the lcm (Lemma 2.1) is elegant and likely to be reusable, and the lower-bound strategy is detailed and appears sound. However, the upper bound as written contains a systematic off-by-one error in counting subsets, which leaves the stated theorem unproved: the argument only yields P_k(x) ≪ x(log x)^{2^k−k−1}, one logarithm too weak for the moment bound. The error appears to be repairable within the manuscript’s scope, but as it stands the central claim is not established.

major comments (1)
  1. [§4, proof of Lemma 4.1] The proof of Lemma 4.1 is only sketched at the point where it is most load-bearing. The uniformity assertion 'Uniformly for numbers u_S ≤ x with min S ≤ ν' is not demonstrated, and the key counting claim that the number of w-tuples modulo d with Δ1 ≡ 0 (mod d) is at most L^{ω(d)} d^{w−1} for a sufficiently large constant L is stated without proof or citation. Since the validity of (4.15) depends entirely on this lemma, please provide the missing details or a reference.
minor comments (5)
  1. [§1, second-to-last paragraph] The sentence 'showing (1.2) can be reduced to computing the quantities P_k(x)' should refer to Theorem 1.1 (or (1.3)), not to (1.2), which is the average-order statement.
  2. [§1, same paragraph] The phrase '(1.4) would imply (1.1)' should read '(1.4) would imply Theorem 1.1', since (1.1) is the well-known asymptotic for the divisor function τ(n).
  3. [§4, after (4.11)] The statement 'there are exactly 2^{k−ν}−1 subsets S ⊆ [k] with min S > ν and S ≠ T0' is incorrect; the correct count is 2^{k−ν}−2. This is the source of the off-by-one in the main argument, so it should be corrected together with the exponents in Lemma 4.1.
  4. [§5, Propositions 5.1 and 5.2] The results in this section depend on Theorem 1.1. Since the upper bound in Theorem 1.1 is not established as written, these propositions are conditional; the dependence should be stated explicitly or the statements should be revised after the upper bound is corrected.
  5. [References] In reference [11], the author name 'Koukoulopolos' is a typo for 'Koukoulopoulos'.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the result is self-contained and confirms, rather than assumes, the Fan-Pomerance conjecture.

full rationale

The paper's central claim (Theorem 1.1) is not an input to the proof. The lower bound (Section 3) starts from the lcm identity of Lemma 2.1, proven in the paper, and then counts admissible prime tuples with Gallagher's theorem, Landau-Page, and Wirsing-type estimates; the upper bound (Section 4) is an induction starting from the known case k=3 only as a base case, and uses an internal decomposition of the lcm together with Brun's sieve, Theorem 4.2 from Pollack, and the auxiliary Lemma 4.1, whose proof is sketched but whose content is an independent estimate rather than a restatement of the target. No fitted parameter is renamed as a prediction, no uniqueness theorem is imported from the author's prior work, and there are no self-citations at all: the references to Fan and Pomerance are to the conjecture being confirmed, not to a premise used in the derivation. The skeptic's off-by-one concern about Lemma 4.1 is a correctness risk, but it is not circularity because the claimed uniformity is not equivalent to Theorem 1.1 and is not obtained by assuming it. The bound (2.1) is therefore derived from stated lemmas and external theorems, not from the conclusion.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

All central estimates are derived from standard results: Landau-Page, Gallagher's PNT, Brun-Titchmarsh, Pollack's upper-bound sieve, Wirsing's mean value theorem, and the known k=2,3 cases. There are no fitted constants in the final asymptotic; the technical constants δ and E are choices that do not affect the order of magnitude. No new entities are postulated.

assumptions (6)
  • domain assumption Landau-Page theorem: there exists an exceptional modulus Q(x), either 1 or a prime, such that no Siegel zero exists for moduli coprime to Q(x) in the stated region (Theorem 3.1).
    Used in Corollary 3.3 to allow lower bounds for primes in arithmetic progressions without Siegel-zero exceptions; cited from [8, Corollary 1].
  • domain assumption Gallagher's prime number theorem: for x ≥ q^D and no exceptional zeros, the number of primes p≤x in a reduced residue class modulo q is ≫ x/(φ(q) log x) (Theorem 3.2).
    Core lower-bound engine for P_k(x); cited from [10] and [13, Lemma 2].
  • standard math Brun-Titchmarsh inequality for primes in arithmetic progressions.
    Used throughout the upper bounds and in Corollary 3.3 and Lemma 4.1; standard sieve theorem.
  • domain assumption Pollack's upper-bound sieve for nonnegative multiplicative functions on sifted sets (Theorem 4.2).
    Used to bound sums over n with nV+1 prime in Section 4; imported from [16, Theorem 1.1].
  • standard math Wirsing's mean value theorem for nonnegative multiplicative functions (Lemma 3.5).
    Used to evaluate the average of g_k(m)/m, producing the (log x)^(2^k-k-1) power; cited from [19] and [11, Exercise 14.6].
  • domain assumption Prior results for k=2 and k=3: ∑ω*^2 ≍ x log x and ∑ω*^3 ≍ x(log x)^4.
    Supplies base cases and consistency checks for the main theorem; from [15], [2], [3], and [6].

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Cite this review

Pith. "Pith review of Moments of the shifted prime divisor function." pith.science (2026). https://pith.science/paper/KCUYXYRQ

@misc{pith2026250524050,
  author       = {Pith},
  title        = {Pith review of: Moments of the shifted prime divisor function},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KCUYXYRQ}},
  note         = {Machine review of arXiv:2505.24050}
}
abstract

Let $\omega^*(n) = \{d|n: d=p-1, \mbox{$p$ is a prime}\}$. We show that, for each integer $k\geq2$, $$ \sum_{n\leq x}\omega^*(n)^k \asymp x(\log x)^{2^k-k-1}, $$ where the implied constant may depend on $k$ only. This confirms a recent conjecture of Fan and Pomerance. Our proof uses a combinatorial identity for the least common multiple, viewed as a multiplicative analogue of the inclusion-exclusion principle, along with analytic tools from number theory.

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Works this paper leans on

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