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REVIEW 3 major objections 5 minor 18 references

The Eighth Power Moments of $\Delta(x)$

T0 review · 3 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read The paper proves that $\int_2^X \Delta(x)^8\,dx$ equals an explicit constant times $X^3$, up to $O(X^{3-1/254+\varepsilon})$.

desk verdict Plausible main term and useful explicit constants, but the key counting lemmas are not rigorous as written; refereeable but not acceptable yet. read the letter →

arxiv 2507.19528 v1 pith:GJ6X76L4 submitted 2025-07-21 math.NT

Junhao Liao , Junjie Liao This is my paper · ORCID
classification math.NT MSC 11L0711B83
keywords divisorproblemerrortermeighthpowermomenttruncatedVoronoiformulaexponentialsumssquare-rootDiophantineequationsasymptoticshortintervals
topics The Collatz 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

The divisor problem asks how the summatory function of divisors deviates from its main term, and $\Delta(x)$ is that deviation. The paper claims that the integral of the eighth power of $\Delta(x)$ from 2 to $X$ has a genuine asymptotic main term proportional to $X^3$, with explicit coefficient $(35C_7-28C_4)/(2048\pi^8)$ and error $O(X^{3-1/254+\varepsilon})$. That would be the first asymptotic formula for the eighth moment, extending known results for the first through fourth powers and fitting the general pattern $\int \Delta^k(x)\,dx \sim C_k X^{1+k/4}$. Such moment formulae turn the hard pointwise question of how large $\Delta(x)$ is into averaged information, with explicit arithmetic constants encoding sums over square-root equalities weighted by divisor functions.

What carries the argument

The load-bearing object is the truncated formula for $\Delta(x)$ as $x^{1/4}$ times a finite sum of terms $d(n)n^{-3/4}\cos(4\pi\sqrt{nx}-\pi/4)$, plus an error term. Expanding the eighth power of this sum produces seven pieces, and the work is to show that six of them are negligible. The negligible pieces are oscillatory integrals controlled by two mechanisms: first-derivative estimates that need a lower bound on nonzero sums of square roots, and counting estimates for near-solutions of equalities between sums of four square roots on each side. Higher-dimensional Weyl differencing enters exactly there, bounding the exponential sums that the smoothed counting argument produces.

What would settle it

Take $k=2$, $N_1=\cdots=N_8=N$, and $\delta=N^{-1}$, $N^{-3/2}$, or $N^{-2}$ in Lemma 13, and enumerate all $n_j\le 2N$ with $0<|\sqrt{n_1}+\cdots+\sqrt{n_4}-\sqrt{n_5}-\cdots-\sqrt{n_8}|<\delta$. If the fitted exponent of the count exceeds the lemma's product bound at accessible $N$, the counting lemma is false and the proof of Lemma 16 loses its error term; a count tracking the claimed power law across these $\delta$ would support the main theorem's exponent.

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Extended reading notes

Core claim

On the paper's own terms, the discovery is Theorem 1: for every fixed integer $X\ge 10$, $$\int_2^X \$\Delta$^8(x)\,dx = \frac{35C_7-28C_4}{2048\$pi^{8}$}\int_2^X $x^{2}$\,dx + O($X^{{3-1/254+\varepsilon}}$),$$ where $C_4$ and $C_7$ are the explicit eight-variable sums in (14) and (15). These sums run over 8-tuples of natural numbers satisfying an equality of sums of square roots, with weights involving $d(n)$ and powers of the variables. The proof first obtains the analogous statement for the truncated sum $\Sigma_Y(x)$ on a dyadic interval $[H,2H]$, controlling all oscillating contributions; only the two non-oscillating matchings survive to produce the constants. Summing dyadic intervals yields the stated asymptotic, and a short-interval version appears as Theorem 2.

Load-bearing premise

The power-saving error rests on Lemma 13's count of near-solutions of a sum-of-square-roots equation, and that lemma's proof delivers a weaker bound than the main argument needs; if the weaker bound fails, the advertised $O(X^{3-1/254+\varepsilon})$ error collapses.

Editorial extensions

If this is right

  • The eighth moment of the divisor error term is of size $X^3$ with a genuine power saving, so the $L^8$ average of $\Delta(x)$ over $[2,X]$ is of order $X^{3/8}$.
  • The leading coefficient is given by the explicit sums in (14) and (15), so the constant in the asymptotic is a concrete arithmetic quantity rather than an unknown parameter.
  • Theorem 2 gives the same main term on short intervals $[X,X+H]$ for $H$ between $X^{7/32+\delta}$ and $X$, with relative error $O(X^{-k})$.
  • The exponent 3 matches the pattern $1+k/4$ at $k=8$, so the result extends the moment pattern previously established for the first four powers.

Reading between the lines

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

  • Going beyond the paper, positivity of the integral forces the constant in Theorem 1 to be positive; checking $35C_7-28C_4>0$ from the two series is a self-contained verification that the main term has the correct sign.
  • A direct numerical test of Lemma 13 with $k=2$ and a range of $\delta$ would show how much of the $1/254$ saving is genuinely available, because the proof of that lemma establishes a weaker bound than the main argument uses.
  • The same route looks extendable to higher even moments, with the square-root counting lemma as the bottleneck; each additional pair of factors adds many more near-equality cases before any new analysis is needed.
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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

3 major / 5 minor

Summary. This paper claims an asymptotic formula for the eighth power moment of the Dirichlet divisor error term: ∫_2^X Δ(x)^8 dx = ((35C7 − 28C4)/(2048π^8)) ∫_2^X x^2 dx + O(X^{3−1/254+ε}), with C4 and C7 defined by explicit eight-variable Diophantine sums. The proof expands the eighth power of Voronoi's truncated formula, identifies diagonal contributions, and bounds off-diagonal oscillatory integrals through a sequence of counting lemmas; a dyadic decomposition yields the stated power saving. A second theorem states a short-interval analogue for H ≥ X^{7/32+δ}.

Significance. The main term is genuinely explicit and is not fitted to data; this is a strength. If the proof were complete, the paper would be the first asymptotic formula for the eighth moment of Δ(x), with a power-saving error, extending the line of work of Tsang, Zhai, and Ivić–Sargos. The claimed error exponent 3−1/254 is also falsifiable and would be a strong test of the conjectured bound Δ(x)≪x^{1/4+ε}. However, the argument currently rests on unverified counting estimates at its core, so the significance can only be assessed conditional on repairing those estimates.

major comments (3)
  1. [§2, Lemma 13] The statement of Lemma 13 is internally inconsistent. The displayed claim has N_j^{1/4}, while the proof concludes with N_j^{1/2}; Lemma 16 in the next section applies the N_j^{1/2} version. This matters because Lemma 13 is the sole estimate behind the off-diagonal terms S5 and S6, which are responsible for the O(H^{3−1/254+ε}) error. In addition, the Fourier-cutoff parameters in the proof cannot satisfy the hypotheses of Lemma 12: setting σ=7δ, the support conditions give b−δ=σ and b+δ=12σ/7, hence b=19σ/14 and δ=5σ/14, violating δ<b/4. The lemma must be repaired before the main theorem is supported.
  2. [§2, Lemma 11] The proof's final estimate uses 1/|∆|≪N^{−1/k+ε}, but for k=2 and h1=h2=h3≈1 the mixed difference ∆ is of size n^{−5/2}, so |∆|^{-1} is ≫N^{5/2−ε}; the asserted upper bound is false in that range. The displayed bound ∫_U^{2U}|S(x,N,k)|^8 dx ≪ (U N^4+N^{8−1/k})N^ε therefore lacks a valid derivation. Since Lemma 13 invokes Lemma 11 in its Hölder step, this invalidates the proof of Lemma 13 independently of the N_j^{1/4} vs N_j^{1/2} issue.
  3. [§2, Lemmas 8–9 and Lemma 16] The counting lemmas for S5/S6 and S2/S3 are not proved in the text. Lemma 8's proof says the bound 'can be found by case analysis' and then gives only fragmentary cases; Lemma 9 says 'the proof is identical', although the equation in Lemma 9 has a different balance of square roots (six terms on one side, one on the other) and is not the same counting problem. Lemma 16 applies these lemmas, together with the unproved Lemma 13, to deduce the H^{3−1/254} bounds for S5 and S6. Thus the error term of Theorem 1 is not supported by the present exposition.
minor comments (5)
  1. [§2, Lemma 11] The proof's notation is inconsistent: after setting H1=N, H2=H^{1/2}, H3=H^{1/4}, it then sums over h1≤N, h2≤N^2, h3≤N^4; also ∂^3f/(∂t1∂t2∂t2) should presumably be ∂t1∂t2∂t3.
  2. [§2, Lemma 8] The function argument has 'S ∼ S' and the parameter list repeats L in A±(N,M,K,L,R,S,L,J); these typos obscure the ranges and should be corrected.
  3. [§2, Lemma 16] The notation 'δ ≍ δL^{1/2}' reuses δ for the threshold and for the small parameter; the two quantities should be renamed to avoid the appearance of a circular definition.
  4. [§2, Lemma 15] As stated ('Let A0>2 be fixed') the lemma claims a bound for all fixed A0; the paper only uses A0=267/27, and the cited result is known only in a restricted range. The statement should be restricted accordingly.
  5. [Throughout] Theorem 1 and Theorem 2 are referred to as Theorem 1.1 and Theorem 1.2, and the abstract says 'the first author' where 'the authors' is meant; there are also name inconsistencies such as 'Dong Guangchang' versus 'Tong K.C.'.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the eighth-power main term is an honest diagonal computation with independently defined constants; the main caveats are rigor gaps in counting lemmas, not circular reasoning.

full rationale

The derivation chain expands the truncated Voronoi sum \Sigma_Y(x)^8 into S1\u2013S7. The main term comes from S4 and S7, which are exact-diagonal sums with coefficients \u22127/32 and 35/128 fixed by the binomial expansion of cos^8. Their integrals are evaluated to \u22127C4/32 \u222bx^2 and 35C7/128 \u222bx^2, where C4 and C7 are defined by the root equations in (14) and (15). This is a direct computation of the leading term, not a fit: the constants are fully specified before the integral is evaluated and are not adjusted to match the target moment. The off-diagonal terms S5/S6 are bounded by counting estimates (Lemmas 8, 13, 14), S2/S3 by analogous lemmas, and the remainder R_{Y,H} is controlled by H\u00f6lder's inequality and the external upper-bound Lemma 15 (Tong) together with standard divisor estimates. There are no fitted parameters renamed as predictions, no load-bearing self-citations (the citations are to Tsang, Ivi\u0107\u2013Sargos, Zhai, Tolev, Kolesnik\u2013Graham, etc., not to prior work of the present authors), and no uniqueness theorem is imported to force a choice. The only circularity-adjacent observation is that the leading constant is by construction the diagonal contribution; that is a standard, non-circular computation because the coefficient is defined independently of the theorem's main term. The rigor defects in Lemma 13 (its proof concludes with N_j^{1/2} rather than the stated N_j^{1/4}, and Lemma 11's reciprocal bound is questionable) are correctness and evidence problems for the error exponent, not circularity: at no point does the proof assume the truth of Theorem 1. Hence the paper is not circular.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

The central claim does not introduce new physical entities or fitted constants. It imports standard results from Voronoi, Tsang, Ivic-Sargos, Kolesnik, Tong, and Tolev. The only hand-chosen parameter is the truncation level Y=H^{11/36}. Because several imported lemmas are stated without proof and one lemma statement conflicts with its proof, the ledger flags these as axioms the reader must accept on trust.

free parameters (1)
  • Y (Voronoi truncation level) = Y = H^{11/36} in Lemma 16; H^epsilon <= Y <= H^{11/36} in Theorem 1
    Truncation point in the Voronoi formula. The claimed error exponent 3-1/254 depends on choosing Y=H^{11/36} to balance the tail bound and the main expansion; no data are fitted, but the proof parameter is hand-chosen.
assumptions (5)
  • standard math Voronoi's truncated formula (13): Delta(x) = x^{1/4}/(pi sqrt(2)) times the sum over n <= N of d(n)n^{-3/4} cos(4 pi sqrt(nx) - pi/4) plus O(X^{1/2+epsilon} N^{-1/2}).
    Starting point of Section 3; the error term is imported from Voronoi [14] without proof.
  • standard math Besicovitch-type lower bound (Lemma 3): any nonzero sum of up to eight signed square roots is bounded below by max(n_i)^{(1-2^{k-1})/2}, in particular max^{-127/2} for eight terms.
    Used repeatedly to separate distinct square-root configurations; proof is cited from Tsang [5].
  • domain assumption Upper-bound moment estimate (Lemma 15): the integral from 1 to T of |Delta(x)|^{A0} dx is O(T^{1+A0/4+epsilon}) for A0 = 267/27.
    Imported from Tong [3]; used to bound the truncated remainder R_{Y,H} in A0 norm via Holder's inequality.
  • standard math Exponential sum lemmas 10 and 17 attributed to 'Kolesnik and Graham [1]', likely meaning Kolesnik [1].
    Used to prove Lemma 11 and Lemma 18; the cited bibliography entry does not actually list a Kolesnik and Graham work.
  • standard math Smooth bump function lemma (Lemma 12) from Tolev [13].
    Used in Lemma 13 to count Diophantine solutions by Fourier transform.

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Pith. "Pith review of The Eighth Power Moments of $\Delta(x)$." pith.science (2026). https://pith.science/paper/GJ6X76L4

@misc{pith2026250719528,
  author       = {Pith},
  title        = {Pith review of: The Eighth Power Moments of $\Delta(x)$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GJ6X76L4}},
  note         = {Machine review of arXiv:2507.19528}
}
abstract

Using Voronoi's truncated formula for $\Delta(x)$ involving Bessel functions, the first author derives an asymptotic formula for the eighth-power moments with an error term of order $O\left(X^{3 - \frac{1}{254} + \varepsilon}\right)$

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

18 extracted references · 17 canonical work pages

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