A note on the $q$-adic valuation of $\sigma_k(n)$

Olivier Bordell\`es

arxiv: 2605.17872 · v1 · pith:PQVRKCZAnew · submitted 2026-05-18 · 🧮 math.NT

A note on the q-adic valuation of σ_k(n)

Olivier Bordell\`es This is my paper

Pith reviewed 2026-05-20 01:01 UTC · model grok-4.3

classification 🧮 math.NT

keywords q-adic valuationsigma_k(n)lifting the exponent lemmacyclotomic polynomialsdivisor sumsarithmetic functionsupper bounds

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The pith

An exact formula for the q-adic valuation of σ_k(n) is derived for odd prime q using the lifting-the-exponent lemma on cyclotomic factors.

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

The paper derives an exact formula for the valuation at an odd prime q of σ_k(n), the sum of the kth powers of the divisors of n. The formula is obtained by applying the lifting-the-exponent lemma to expressions built from cyclotomic polynomials that arise when writing σ_k(n) in terms of the divisors. A reader would care because the exact formula immediately supplies an explicit upper bound on this valuation that is asymptotically tighter than Zhao's earlier bound, at least when n is large and k is at least q minus 2. The result gives sharper control over the highest power of q that can divide these divisor sums for many integers n.

Core claim

We obtain an exact formula for the q-adic valuation of σ_k(n) where q is an odd prime, allowing us to derive an explicit upper bound which is asymptotically better than the previous bound obtained by Zhao when n is large and k ≥ q-2. The key parts are played by the LTE lemma and the use of cyclotomic polynomials.

What carries the argument

Lifting-the-exponent lemma applied directly to cyclotomic polynomial expressions arising from the divisors of n.

If this is right

An explicit upper bound on the q-adic valuation of σ_k(n) that is asymptotically better than Zhao's bound for large n when k ≥ q-2.
Precise determination of the exact power of q dividing σ_k(n) for n and k satisfying the conditions.
Sharper estimates on the size of σ_k(n) in the q-adic sense for use in bounding related arithmetic quantities.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same cyclotomic-plus-LTE approach could be tested on valuations of other divisor sums such as the ordinary sum-of-divisors function for k=1.
Numerical checks on sequences of n with fixed q and increasing size would directly confirm the claimed improvement in the upper bound.
The method may adapt to give similar exact formulas when the prime q is replaced by 2, provided a suitable version of the lifting lemma is available.

Load-bearing premise

The lifting-the-exponent lemma applies directly to the relevant polynomial expressions arising from the divisors of n under the stated conditions on k and q, with no additional terms missed by the cyclotomic factorization.

What would settle it

Pick a specific large n divisible by an odd prime q with k at least q-2, compute the actual sum of kth powers of its divisors, factor out the exact power of q dividing that sum, and check whether the value matches the claimed exact formula or stays below the new upper bound.

read the original abstract

In this note, we obtain an exact formula for the $q$-adic valuation of $\sigma_k(n)$ where $q$ is an odd prime, allowing us to derive an explicit upper bound which is asymptotically better than the previous bound obtained by Zhao when $n$ is large and $k \geqslant q-2$. The key parts are played by the LTE lemma and the use of cyclotomic polynomials.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This note gives an exact formula for v_q(σ_k(n)) via LTE on cyclotomics that improves Zhao's asymptotic bound for large n and k ≥ q-2.

read the letter

This note gives an exact formula for the q-adic valuation of σ_k(n) when q is an odd prime and k ≥ q-2. From there it extracts an upper bound that is asymptotically tighter than Zhao's for large n. The derivation rests on factoring the divisor sum through cyclotomic polynomials and feeding the pieces into the lifting-the-exponent lemma under the stated conditions on k. That produces a closed-form valuation rather than just an inequality, which is the concrete advance over the cited prior work. The setup is standard and the citation pattern is appropriate, pointing back to Zhao without circularity or extra assumptions. The approach looks honest and directly applicable to the narrow question at hand. One place worth a closer look is whether multiple cyclotomic factors or divisor terms can share the same minimal q-valuation. If they do, their sum could cancel and raise the total valuation, which would change both the exact formula and the claimed improvement in the bound. The stress-test note flags this possibility, and the paper needs to confirm that the conditions on k isolate a unique minimal term or that all cases are covered without extra cancellation. A referee can settle this quickly from the full proof. The paper is aimed at specialists already working on p-adic valuations of arithmetic functions. Most number theorists will find it too narrow to change their own work, but someone needing a sharper explicit bound for specific k and q will get direct use from it. The thinking is clear and the tools are applied without fitting or invented steps. I would send it to peer review for a short check on the case analysis and the cancellation question rather than desk-reject it.

Referee Report

1 major / 1 minor

Summary. The manuscript claims an exact formula for the q-adic valuation v_q(σ_k(n)) when q is an odd prime, obtained by expressing σ_k(n) via cyclotomic polynomials and applying the lifting-the-exponent lemma under the hypothesis k ≥ q-2. From this formula the authors derive an explicit upper bound that is asymptotically sharper than Zhao's bound for large n.

Significance. If the exact formula is valid, the work supplies a sharper arithmetic tool for controlling the q-adic size of divisor power sums, which could be useful in questions about the distribution of σ_k(n) modulo powers of q. The improvement over the earlier bound is concrete and depends directly on the exactness of the valuation formula.

major comments (1)

[Main derivation] Main derivation (the paragraph following the statement of the exact formula): the claim that LTE applied to the individual cyclotomic factors Φ_d(n) yields the precise minimal valuation of the sum requires an explicit argument that no two or more terms attain the same minimal q-valuation and cancel. The manuscript does not appear to rule out such cancellation under the stated conditions on k and q; if cancellation occurs the exact formula would fail and the subsequent asymptotic bound would need adjustment.

minor comments (1)

[Theorem statement] The statement of the exact formula should include the precise range of n for which it holds (e.g., whether n is required to be coprime to q or not).

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and the constructive comment on the main derivation. We address the concern regarding potential cancellations below.

read point-by-point responses

Referee: Main derivation (the paragraph following the statement of the exact formula): the claim that LTE applied to the individual cyclotomic factors Φ_d(n) yields the precise minimal valuation of the sum requires an explicit argument that no two or more terms attain the same minimal q-valuation and cancel. The manuscript does not appear to rule out such cancellation under the stated conditions on k and q; if cancellation occurs the exact formula would fail and the subsequent asymptotic bound would need adjustment.

Authors: We thank the referee for highlighting this point. We agree that an explicit argument is required to confirm that the minimal q-valuation of the sum is indeed attained without cancellation among the cyclotomic terms. In the revised manuscript we will add a short lemma immediately after the exact formula, showing that under k ≥ q-2 the valuations v_q(Φ_d(n)) for the relevant d are strictly ordered or that any collection of minimal-valuation terms sums to a nonzero residue modulo q^{m+1} by the explicit form of the cyclotomic polynomials and the LTE lifting conditions. This addition will substantiate the claimed exact formula and the resulting asymptotic bound. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation uses independent external lemmas and classical factorizations

full rationale

The paper obtains its exact formula for v_q(σ_k(n)) by rewriting the divisor sum in terms of cyclotomic polynomials and then applying the standard lifting-the-exponent lemma to the resulting terms when k ≥ q-2. Both the cyclotomic factorization and LTE are classical, externally verified tools whose statements and proofs do not depend on the present work or on any fitted parameters from it. No step equates a claimed prediction to a fitted input by construction, renames a known result, or rests on a self-citation chain that itself lacks independent verification. The derivation therefore remains self-contained against external mathematical benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the applicability of the lifting-the-exponent lemma and the completeness of the cyclotomic factorization for the divisor sum; no free parameters or new entities are introduced.

axioms (2)

domain assumption Lifting The Exponent lemma applies to the polynomial expressions generated by the divisors of n under the given constraints on k and q
Invoked as the key tool for obtaining the exact valuation formula.
domain assumption Cyclotomic polynomials capture all prime-power contributions to σ_k(n) without residual terms
Used to factor the relevant sums and isolate the q-power.

pith-pipeline@v0.9.0 · 5582 in / 1293 out tokens · 72338 ms · 2026-05-20T01:01:41.814078+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 1 … v_q(σ_k(n)) = ∑_{p^α∥n, p^k≡1(mod q)} v_q(α+1) + ∑_{p^α∥n, p^k≢1(mod q)} 1_{ϖ_k | α+1} (v_q(α+1)+v_q(k)+v_q(Φ_ϖ1(p)))
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Lemma 4 (LTE) … v_q(a^m - b^m) = v_q(a-b) + v_q(m)

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

10 extracted references · 10 canonical work pages

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K. CHENG ANDK. ZHANG, On the 2-adic valuation ofσ k(n), preprint 8 pp., 2026, arXiv:2603.11979

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AMDEBERHAN, V

T. AMDEBERHAN, V. H. MOLL, V. SHARMA,ANDD. VILLAMIZAR, Arithmetic properties of the sum of divisors,J. Number Theory223(2021), 325–349

work page 2021

[2] [2]

BORDELLÈS,Arithmetic Tales, Advanced Edition, UTX, Springer, 2020

O. BORDELLÈS,Arithmetic Tales, Advanced Edition, UTX, Springer, 2020

work page 2020

[3] [3]

CHENG ANDK

K. CHENG ANDK. ZHANG, On the 2-adic valuation ofσ k(n), preprint 8 pp., 2026, arXiv:2603.11979

work page arXiv 2026

[4] [4]

LUCAS, Théorie des fonctions numériques simplement périodiques [Continued],Amer

E. LUCAS, Théorie des fonctions numériques simplement périodiques [Continued],Amer . J. Math.1 (1878), 197–240

work page

[5] [5]

MANEA, Somea n ±b n problems in number theory,Math

M. MANEA, Somea n ±b n problems in number theory,Math. Mag.79(2) (2006), 140–145

work page 2006

[6] [6]

NICOLAS ANDG

J.-L. NICOLAS ANDG. ROBIN, Majorations explicites pour le nombre de diviseurs den,Canad. Math. Bull.39(1983), 485–492

work page 1983

[7] [7]

ROBIN, Estimation de la fonction de Tchebychefθsur lek-ième nombre premier et grandes valeurs de la fonctionω(n) nombre de diviseurs premiers den,Acta Arith.42(1983), 367–389

G. ROBIN, Estimation de la fonction de Tchebychefθsur lek-ième nombre premier et grandes valeurs de la fonctionω(n) nombre de diviseurs premiers den,Acta Arith.42(1983), 367–389

work page 1983

[8] [8]

L. C. WASHINGTON,Introduction to Cyclotomic Fields, 2nd Ed., GTM 83, Springer, 1997

work page 1997

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ZHAO ANDY

J. ZHAO ANDY. CHEN,p-adic valuation of the sum of divisors,Front. Math.20(2025), 795–827

work page 2025

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ZHAO,p-adic valuation ofσ k(n),Bull

J. ZHAO,p-adic valuation ofσ k(n),Bull. Aust. Math. Soc.(2026), 1–5. 5

work page 2026