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Moments of the number of representations as sums of two prime squares

T0 review · 0 major / 6 minor · reviewed 2026-07-13 · grok-4.5

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Pith's one-line read The kth moments of representations as sums of two prime squares have the expected size for every fixed k≥4, unconditionally.

desk verdict Unconditional order-of-magnitude for all higher moments of r₂, via a clean mixed-moment lower bound and a friable refinement that kills the last logloglog. read the letter →

arxiv 2607.08985 v1 pith:AXCPN4C2 submitted 2026-07-09 math.NT

Haozhe Gou This is my paper · ORCID
classification math.NT MSC 11N3711P3211A25
keywords primesquaresrepresentationfunctionsmomentsSelbergsievefriableintegersshifteddivisorfunction
topics Goldbach's Conjecture
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

How often can an integer be written as a sum of two prime squares, and how large do those representation counts get on average? This paper shows that the kth power of the representation function r2, summed up to x, has the correct order of magnitude x times a precise power of log x, for every fixed k at least 4. The matching lower bounds no longer need a conjectural uniform Green–Tao statement about primes in linear systems; instead one keeps a single prime-square representation and supplies the remaining multiplicity from the classical sum-of-two-squares function r0, which is accessible by primes in arithmetic progressions on average. On the upper side the same Gaussian factorisation and Selberg sieve as earlier work are retained, but a friable Selberg–Delange estimate is inserted in the medium-prime range so that a Dickman decay factor removes the extra log-log-log that previously spoiled the fourth moment. The same mixed-moment idea also yields a short unconditional proof of the lower bounds for moments of the shifted-prime divisor function.

What carries the argument

The mixed moment Mk(x)=∑ R2(n)r0(n)^{k-1}. Hölder together with the known moments of r0 converts a lower bound for Mk into a lower bound for the pure moments of R2; the mixed count reduces to primes satisfying a single congruence m|p^{2}+q^{2}, which is handled by Bombieri–Vinogradov.

What would settle it

If an explicit computation of the fourth moment up to a large x (say 10^{12}) produced a growth rate visibly larger than x/(log x)^5 by more than a slowly growing log-log-log factor, the claimed upper bound would be false.

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Extended reading notes

Core claim

For every fixed integer k≥4 the sum of r2(n)^k over n≤x is asymptotic in order of magnitude to x(log x)^{2^{k-1}-2k-1}. The upper bound for k=4 loses the previous log-log-log factor, the third-moment error reaches the conjecturally optimal O(x/(log x)^3), and all lower bounds hold unconditionally.

Load-bearing premise

The medium-range upper bound needs a positive-power divisor saving on the singular series over friable bases; if that saving vanished, the Dickman decay alone would not cancel the remaining harmonic sum for k=4.

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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

0 major / 6 minor

Summary. The paper establishes the correct order of magnitude for the kth moments of r2(n), the number of representations of n as a sum of two prime squares, for every fixed integer k≥4: ∑_{n≤x} r2(n)^k ≃_k x (log x)^{2^{k-1}-2k-1}. After the elementary reduction (2.1) to the ordered counting function R2, the upper bound is obtained from factorial moments Sk(x) (Proposition 2.1) via Sabuncu’s Gaussian factorization and Selberg-sieve framework, with a new friable Selberg–Delange input (Tenenbaum–Wu) that inserts a Dickman-type factor into the medium-prime range and removes the previous log log log x loss for k=4. The matching lower bound is obtained unconditionally by replacing the pure moment of R2 by the mixed moment Mk(x)=∑ R2(n) r0(n)^{k-1}, applying Hölder, and estimating the mixed moment via the character expansion of r0 together with Bombieri–Vinogradov averages for prime pairs (Lemmas 3.1–3.3, §7). As a corollary one obtains the optimal error term for the third moment of r2. The same mixed-moment method yields a simpler unconditional proof of the lower bounds for the moments of the shifted-prime divisor function ω*, recovering the lower-bound half of Gabdullin’s resolution of the Fan–Pomerance conjecture.

Significance. The result closes the remaining gap left by Sabuncu for the moments of r2: the upper bound for k=4 is now of the expected order, and all lower bounds for k≥4 are unconditional. The mixed-moment reduction (keeping one prime-square representation and supplying the rest from r0) is a clean, reusable idea that avoids any appeal to a uniform Green–Tao theorem and immediately extends to ω* and, as the author indicates, to r1. The upper-bound improvement is likewise concrete: the friable average of the singular series (Proposition 5.1) produces a square-root Dickman factor that is more than enough to absorb the previous logarithmic loss. Both pillars rest on classical tools (Bombieri–Vinogradov, Selberg sieve, Tenenbaum–Wu) applied inside their standard ranges, so the paper supplies a definitive unconditional statement of the expected order of magnitude for these moments.

minor comments (6)
  1. [Abstract / Bibliography] In the abstract and introduction the arXiv identifier of Sabuncu is written [Sabuncu2024] while the bibliography uses [Sab24]; unify the citation key.
  2. [Section 2.1] Section 2.1, display after (2.2): the exponent arithmetic that converts Mk(x) ≫ x (log x)^{2^{k-1}-3} into the pure-moment lower bound is correct but written in a single dense line; a short intermediate step would help the reader.
  3. [Lemma 3.2] Lemma 3.2: the application of Cauchy–Schwarz followed by the trivial bound E(m) ≪ P log log(m+2)/m is standard, but the final choice of A relative to B is left implicit; a one-line remark that A can be taken larger than any fixed power of B would make the dependence transparent.
  4. [Proposition 5.1] Proposition 5.1: the geometric-mean step that produces ho_{2^{k-1}}(u)^{1/2} is legitimate for upper bounds, yet a brief sentence noting that any positive power of the divisor saving would suffice would clarify the robustness of the argument.
  5. [Section 8] Section 8, display (8.6): the local factor g_k(ℓ) is written correctly, but the subsequent appeal to Selberg–Delange for ∑ g_k(c)/φ(c) would benefit from an explicit reference to the same argument used in Lemma 3.3.
  6. A few typographical inconsistencies appear (e.g., “Erdös” vs. “Erdős”, occasional missing spaces around ≃ and ≪). A light copy-edit pass would remove them.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: moments derived from classical sieve/AP tools and mixed-moment Hölder, not from self-defined or fitted targets.

full rationale

The paper’s load-bearing chain is classical and non-circular. The lower bound replaces the pure moment of R2 by the mixed moment Mk = ∑ R2 r0^{k−1}, applies Hölder with the known Blomer–Granville moments of r0, and evaluates Mk via Bombieri–Vinogradov averages and a Selberg–Delange Euler product over squarefree m ≡ 1 (mod 4); none of these quantities is defined in terms of the target moment. The upper bound follows Sabuncu’s Selberg-sieve / largest-prime-factor decomposition but inserts an independent friable Selberg–Delange estimate (Tenenbaum–Wu) to retain a Dickman factor in the medium range; the geometric-mean step that produces ρ^{1/2} is a legitimate upper-bound device, not a fit or a self-definition. Citations of Sabuncu supply intermediate combinatorial and sieve lemmas that are restated with explicit ranges and are not the claimed asymptotic; the Green–Tao-type conjecture that Sabuncu needed for lower bounds is avoided entirely. The ω* application is the same mixed-moment template with τ, again independent of the target. No quantity is fitted to data and re-presented as a prediction, no uniqueness theorem is imported from the author’s own prior work, and no ansatz is smuggled in via self-citation. Score 0 is therefore appropriate.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The paper works entirely inside classical analytic number theory. The load-bearing external inputs are standard theorems (prime-number theorem, Bombieri–Vinogradov, Selberg sieve, Selberg–Delange method, friable estimates of Tenenbaum–Wu, known moments of r0). No free parameters are fitted and no new physical or arithmetic entities are postulated.

assumptions (4)
  • standard math Bombieri–Vinogradov theorem in the form needed for the weighted error sum of Lemma 3.2 (level of distribution 1/3 with arbitrary logarithmic power).
    Used to control the error term after expanding the mixed moment Mk(x); without it the lower bound would remain conditional.
  • standard math Friable Selberg–Delange theorem of Tenenbaum–Wu (Theorem 4.1 / Corollaire 2.3 of [TW03]) for multiplicative functions with the stated prime-power and prime-sum hypotheses.
    Supplies the Dickman factor ρ_{2^{k-1}}(u) that removes the logloglog loss in the medium range.
  • standard math Selberg upper-bound sieve applied to the 2k+1 linear and quadratic forms after Gaussian factorization (Lemma 5.2, taken from Sabuncu).
    Converts the counting problem for each fixed base tuple (m⃗,n⃗) into a singular-series times z/(log z)^{2k+1}.
  • standard math Asymptotic formula for the moments of the classical representation function r0(n) (Blomer–Granville).
    Used via Hölder to convert the mixed-moment lower bound into a pure-moment lower bound.

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Pith. "Pith review of Moments of the number of representations as sums of two prime squares." pith.science (2026). https://pith.science/paper/AXCPN4C2

@misc{pith2026260708985,
  author       = {Pith},
  title        = {Pith review of: Moments of the number of representations as sums of two prime squares},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AXCPN4C2}},
  note         = {Machine review of arXiv:2607.08985}
}
abstract

We prove, for every fixed integer $k\ge 4$, the correct order of magnitude for the $k$th moments of the function that counts the number of representations of an integer as sums of two prime squares. The upper bound for $k=4$ was previously known up to $\log\log\log x$, and the lower bound for $k\ge 4$ was only known conditionally on a conjectural uniform version of the Green-Tao theorem on linear equations in primes by the work of Sabuncu \cite{Sabuncu2024}. As an application of our method, we give a simpler proof of the lower bounds for the moments of the shifted prime divisor function, thereby recovering the lower-bound part of Gabdullin's recent result on a conjecture of Fan and Pomerance.

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Reference graph

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