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arxiv: 2606.25849 · v1 · pith:ZA6LU6VQnew · submitted 2026-06-24 · 🧮 math.NT

A resolution of ErdH{o}s Problem 1061 on the sum-of-divisors function

Eric Li (Trinity College , University of Cambridge) This is my paper

Pith reviewed 2026-06-25 20:06 UTC · model grok-4.3

classification 🧮 math.NT
keywords sum-of-divisors functionErdős problemsabundancy indexprime tuplesSiegel-Walfisz theoremquadric surfacesparameter sieve
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The pith

S(x) tends to infinity faster than x times any fixed power of log x.

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

The paper resolves Erdős Problem 1061 by proving that the count S(x) of pairs (a,b) with a+b ≤ x and σ(a)+σ(b)=σ(a+b) satisfies lim S(x)/(x (log x)^R) = +∞ for every R>0. It constructs these pairs by starting with three integers of equal abundancy index, reducing the identity to a pair of equations in six primes that lie on a split quadric, and then using a three-parameter rational ruling to produce many six-tuples of linear forms. An exact lattice-index calculation combined with a codimension-two parameter sieve and Bienvenu's higher-dimensional Siegel-Walfisz theorem produces sufficiently many prime points on these forms, after which coprime multiplier amplification yields the growth statement.

Core claim

For every R>0, lim_{x→∞} S(x)/(x (log x)^R) = +∞, where S(x) counts the ordered pairs (a,b) of natural numbers satisfying a+b≤x and σ(a)+σ(b)=σ(a+b).

What carries the argument

The three-parameter rational ruling of the split quadric that produces the affine systems of six linear forms on which the prime points are located.

If this is right

  • The divisor-sum identity σ(a)+σ(b)=σ(a+b) holds for infinitely many pairs at every logarithmic scale.
  • Abundancy-index equality can be realized by integers whose prime factors come from arbitrarily many distinct linear forms in six variables.
  • The set of solutions is dense enough that S(x) exceeds every polynomial in log x multiplied by x.
  • The method produces solutions in which a, b, and a+b are all highly composite in a controlled way.

Where Pith is reading between the lines

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

  • The same quadric-ruling technique may apply to other additive problems involving the sum-of-divisors function on multiple variables.
  • If the sieve can be strengthened, one might obtain an asymptotic for S(x) with a slowly growing factor such as (log log log x)^c.
  • The construction suggests that the abundancy index takes the same value on three integers whose sum is controlled, which could be tested numerically for small primes.

Load-bearing premise

The codimension-two parameter sieve together with Bienvenu's higher-dimensional Siegel-Walfisz theorem supplies sufficiently many prime points uniformly on the affine systems of six linear forms obtained from the three-parameter rational ruling of the split quadric.

What would settle it

An upper bound S(x) ≤ C x (log x)^K for some fixed C and K and all large x would falsify the claim.

read the original abstract

We resolve Erd\H{o}s Problem 1061, the question whether the number \[ S(x)=\#\{(a,b)\in\mathbb{N}^2:a+b\le x, \ \sigma(a)+\sigma(b)=\sigma(a+b)\} \] of ordered solutions has a linear asymptotic $S(x)\sim cx$. In fact the opposite extreme holds at every fixed logarithmic scale: for every \(R>0\), \[ \lim_{x\to\infty}\frac{S(x)}{x(\log x)^R}=+\infty. \] The construction begins with three integers having the same abundancy index and reduces the divisor-sum identity to two equations in six primes. After a linear change of variables, these equations lie on a split quadric. A three-parameter rational ruling of the quadric supplies many affine systems of six linear forms. An exact lattice-index calculation, an elementary codimension-two parameter sieve, and Bienvenu's higher-dimensional Siegel--Walfisz theorem give prime points uniformly on these planes. Coprime multiplier amplification then yields the stated resolution.

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.

Referee Report

2 major / 2 minor

Summary. The manuscript resolves Erdős Problem 1061 by proving that S(x), the number of ordered pairs (a,b) ∈ ℕ² with a+b ≤ x and σ(a)+σ(b)=σ(a+b), satisfies lim_{x→∞} S(x)/(x (log x)^R) = +∞ for every fixed R>0. The argument begins with triples of equal abundancy index, reduces the identity to two equations in six primes, applies a linear change of variables to place them on a split quadric, uses a three-parameter rational ruling to produce affine systems of six linear forms, computes the exact lattice index, and invokes an elementary codimension-two parameter sieve together with Bienvenu's higher-dimensional Siegel--Walfisz theorem to locate sufficiently many prime points, followed by coprime multiplier amplification.

Significance. If the uniformity statements hold, the result supplies a strong resolution showing superpolylogarithmic growth of S(x) at every fixed scale. The manuscript is notable for its exact lattice-index calculation (an unconditional algebraic step) and for combining a rational parametrization of the quadric with an external analytic sieve theorem; these features would constitute a genuine contribution to the arithmetic of the sum-of-divisors function.

major comments (2)
  1. [Abstract, final paragraph] Abstract, final paragraph: the claim that Bienvenu's higher-dimensional Siegel--Walfisz theorem supplies sufficiently many prime points uniformly on the three-parameter family of affine systems requires an explicit uniformity statement. The coefficients of the six linear forms grow with x (via the ruling parameters), and it is not clear whether the theorem's error terms remain controllable in this regime without additional losses that would prevent the lower bound from holding for arbitrarily large R.
  2. [Reduction step (described in abstract)] Reduction step (described in abstract): after the linear change of variables that places the divisor-sum equations on the split quadric, the codimension-two parameter sieve is applied to the resulting six linear forms. The manuscript must verify that the lattice-index calculation produces forms whose content and gcd properties are compatible with the sieve hypotheses uniformly over the three-parameter family; any x-dependent loss here would be load-bearing for the claimed growth rate.
minor comments (2)
  1. The definition of S(x) is given in the abstract but should be restated verbatim in the introduction for self-contained reading.
  2. All citations to Bienvenu's theorem should include the precise statement or theorem number used, together with any uniformity hypotheses invoked.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and for identifying points where additional explicit verification would strengthen the manuscript. We address each major comment below and will incorporate the requested clarifications in a revised version.

read point-by-point responses

  1. Referee: [Abstract, final paragraph] Abstract, final paragraph: the claim that Bienvenu's higher-dimensional Siegel--Walfisz theorem supplies sufficiently many prime points uniformly on the three-parameter family of affine systems requires an explicit uniformity statement. The coefficients of the six linear forms grow with x (via the ruling parameters), and it is not clear whether the theorem's error terms remain controllable in this regime without additional losses that would prevent the lower bound from holding for arbitrarily large R.

    Authors: We agree that an explicit uniformity statement is required. The coefficients of the six linear forms are bounded by a fixed power of the ruling parameters (which are at most x^ε for arbitrarily small ε in the range used for the lower bound). Bienvenu's theorem supplies the necessary uniformity in this regime, with error terms that do not introduce R-dependent losses. In the revision we will add a dedicated paragraph (or short subsection) spelling out the coefficient growth, the applicable range of the theorem, and the resulting error control, thereby making the uniformity fully explicit. revision: yes

  2. Referee: [Reduction step (described in abstract)] Reduction step (described in abstract): after the linear change of variables that places the divisor-sum equations on the split quadric, the codimension-two parameter sieve is applied to the resulting six linear forms. The manuscript must verify that the lattice-index calculation produces forms whose content and gcd properties are compatible with the sieve hypotheses uniformly over the three-parameter family; any x-dependent loss here would be load-bearing for the claimed growth rate.

    Authors: The lattice index is computed exactly in Section 4 and equals a fixed positive integer independent of the ruling parameters and of x. The six linear forms produced by the change of variables are primitive (content 1) by construction of the rational ruling, and the gcd conditions required by the codimension-two sieve hold uniformly because the transformation matrix is integral with determinant ±1. We will insert a short remark immediately after the index calculation confirming that these properties are uniform over the three-parameter family and introduce no x-dependent factors. revision: yes

Circularity Check

0 steps flagged

No circularity: external theorems and algebraic reductions supply the bound

full rationale

The derivation begins with abundancy-index triples, reduces the divisor-sum condition to equations on a split quadric, applies an exact lattice-index computation, then invokes an elementary codimension-two sieve together with Bienvenu's external higher-dimensional Siegel-Walfisz theorem to locate prime points uniformly over the three-parameter ruling. None of these steps defines the target asymptotic S(x) in terms of itself, fits a parameter to a subset of the same data, or relies on a self-citation chain whose cited result is itself unverified. The uniformity claim is an application of cited external analytic number theory rather than a renaming or self-referential construction. The paper is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The construction relies on standard facts about the sum-of-divisors function and external analytic results rather than introducing new free parameters or invented entities.

axioms (2)
  • domain assumption The sum-of-divisors function σ is multiplicative
    Used implicitly when working with abundancy indices and reducing the identity.
  • standard math Split quadrics over the rationals admit three-parameter rational rulings
    Invoked to produce the affine systems of linear forms.

pith-pipeline@v0.9.1-grok · 5726 in / 1350 out tokens · 28327 ms · 2026-06-25T20:06:34.772530+00:00 · methodology

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Reference graph

Works this paper leans on

8 extracted references · 5 canonical work pages

  1. [1]

    Bienvenu,A higher-dimensional Siegel–Walfisz theorem, Acta Arith.179(2017), no

    P.-Y. Bienvenu,A higher-dimensional Siegel–Walfisz theorem, Acta Arith.179(2017), no. 1, 79–100.https: //doi.org/10.4064/aa8600-10-2016

    work page doi:10.4064/aa8600-10-2016 2017

  2. [2]

    T. F. Bloom,Erdős Problem #1061, Erdős Problems database, accessed June 24, 2026. https://www. erdosproblems.com/1061

    2026

  3. [3]

    Green and T

    B. Green and T. Tao,Linear equations in primes, Ann. of Math. (2)171(2010), no. 3, 1753–1850.https: //doi.org/10.4007/annals.2010.171.1753. RESOLUTION OF ERDŐS PROBLEM 1061 25

    work page doi:10.4007/annals.2010.171.1753 2010

  4. [4]

    Green and T

    B. Green and T. Tao,The Möbius function is strongly orthogonal to nilsequences, Ann. of Math. (2)175(2012), no. 2, 541–566.https://doi.org/10.4007/annals.2012.175.2.3

    work page doi:10.4007/annals.2012.175.2.3 2012

  5. [5]

    Green, T

    B. Green, T. Tao and T. Ziegler,An inverse theorem for the GowersU s+1[N ]-norm, Ann. of Math. (2)176 (2012), no. 2, 1231–1372.https://doi.org/10.4007/annals.2012.176.2.11

    work page doi:10.4007/annals.2012.176.2.11 2012

  6. [6]

    https://oeis.org/A110177

    OEIS Foundation Inc.,The On-Line Encyclopedia of Integer Sequences, A110177, accessed June 24, 2026. https://oeis.org/A110177

    2026

  7. [7]

    Pollack and C

    P. Pollack and C. Pomerance,Some problems of Erdős on the sum-of-divisors function, Trans. Amer. Math. Soc. Ser. B3(2016), 1–26.https://doi.org/10.1090/btran/10

    work page doi:10.1090/btran/10 2016

  8. [8]

    R. K. Guy,Unsolved Problems in Number Theory, 3rd ed., Problem Books in Mathematics, Springer, New York, 2004; Problem B15

    2004