REVIEW 3 major objections 5 minor 6 references
Moment estimates for the exponential sum with higher divisor functions
T0 review · 3 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash
Pith's one-line read The s-th moments of divisor sums now have a full asymptotic
desk verdict New and useful moment asymptotics for τ_k sums, but the major-arc error term in Proposition 2.1 is asserted, not proved — a fixable gap, not a fatal one. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The load-bearing identity is the circle-method factorization near a rational. With $P=X^{\eta}$, $\eta=2\delta_{s,k}/(s-2)$, the interval is split into major arcs $|\alpha-a/q|\le P/X$ for $q\le P$, and on such an arc $e(n(a/q+\beta))=e(na/q)e(n\beta)$. The sum over $n$ therefore factors into an additive-twist estimate for $\tau_k$ (a polynomial in $\log X$ plus an explicit error) multiplied by the Dirichlet kernel $v(\beta)=\sum_{n\le X}e(n\beta)$, whose $s$-th moment is handled separately: $\int_0^1|v(\beta)|^s\,d\beta = A_s X^{s-1}+O(X^{s-2})$. On the minor arcs, $\tau_k$ is decomposed into type I and type II convolutions to bound the sum pointwise by $\ll X^{1-\eta/2}(\log X)^{O(1)}$, and the exact mean-square identity converts that supremum into the claimed power saving.
What would settle it
Evaluate $S_q(a)=\sum_{n\le X}\tau_k(n)e(an/q)$ numerically for one fixed $k\ge 2$ and for $q$ in the range used in the major arcs, and compare the size of $S_q(a)-XP_{k,q}(\log X)$ with $q^{1/2+k/(k+1)}X^{(k-1)/(k+1)}(qX)^\varepsilon$; an exponent visibly larger than the stated one would falsify the main theorem. Alternatively, compute $\int_0^1 |M(\alpha)|^s\,d\alpha$ for a single pair $(s,k)$ and check that the deviation from the leading asymptotic decays like $X^{-\delta_{s,k}}$ rather than like $X^{-c}$ for a smaller $c$.
Extended reading notes
Core claim
The central claim is that for $k\ge 2$ and $s>2$, $$\$int_0^{1}$ \Bigl|\sum_{n\le X}\tau_k(n)e(n\$\alpha$)\Bigr|^s\,d\$\alpha$ = $X^{{s-1}}$(\log X)^{s(k-1)}\sum_{\ell\ge 0}\frac{\gamma_{\ell,s,k}}{(\log X)^\ell} + O\bigl($X^{{s-1-\delta_{s,k}}$+\varepsilon}\bigr),$$ where $\delta_{s,k}=2(s-2)/((s+7)(k+1)+2)$ and $\gamma_{0,s,k}>0$. Thus the moment has a known positive main term of size $X^{s-1}(\log X)^{s(k-1)}$, and the error is genuinely smaller by a power of $X$. The proof establishes the main term from the major arcs and shows the minor arcs contribute no larger error, verifying the prediction that rationals with small denominator dominate such moments.
Load-bearing premise
The argument assumes, without proving it in this paper, that the additive-twist sum $\sum_{n\le X}\tau_k(n)e(an/q)$ has main term $X$ times a degree $k-1$ polynomial in $\log X$ with error $O(q^{1/2+k/(k+1)}X^{(k-1)/(k+1)}(qX)^\varepsilon)$; a worse error would make the advertised power saving in Theorem 1.1 collapse.
Editorial extensions
If this is right
- For each fixed $s>2$, the $L^s$ norm of $M$ is asymptotic to $C_{s,k}^{1/s}X^{1-1/s}(\log X)^{k-1}$, so the true order of magnitude is no longer open.
- The major arcs alone produce the main term up to the stated power saving, which settles that the moment problem for $\tau_k$ has no substantial secondary contribution from intermediate rationals.
- The asymptotic expansion $\sum_{\ell\ge 0}\gamma_{\ell,s,k}/(\log X)^\ell$ provides arbitrarily high-order approximations with coefficients bounded by $\exp(O(\ell))$, so truncating it gives effective estimates whose error is smaller by powers of $\log X$ as well as by a power of $X$.
- Combined with the minor-arc pointwise bound, the theorem yields quantitative control on the measure of the set where $|M(\alpha)|$ is large, in the parameter ranges used by applications of the circle method.
Reading between the lines
- The same proof should transfer to sums of $k$ fixed Dirichlet characters, with the same $\delta_{s,k}$ and only the shape of the polynomial $P_{k,q}$ changing, since the additive-twist and type I/II ingredients carry over.
- A stronger additive-twist error term, if available, would push the power saving $\delta_{s,k}$ toward the natural barrier $1/k$; the author flags that beating this barrier qualitatively is likely hard.
- For higher-rank cusp-form coefficients, the major-arc-dominance mechanism is expected to fail because those exponential sums are small everywhere; their moments would require a completely different method.
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper studies the L^s mean of the exponential sum M(α)=∑_{n≤X}τ_k(n)e(nα) for fixed integer k≥2 and real s>2. Theorem 1.1 claims an asymptotic of the form X^{s-1}(log X)^{s(k-1)}∑_{ℓ≥0}γ_{ℓ,s,k}(log X)^{-ℓ}+O(X^{s-1-δ_{s,k}+ε}) with an explicit power saving δ_{s,k}=2(s-2)/((s+7)(k+1)+2), coefficients satisfying |γ_{ℓ,s,k}|≪exp(O(ℓ)), and positive leading coefficient γ_{0,s,k}>0. The proof splits [0,1] into major arcs near rationals a/q with q≤P=X^η and minor arcs. The minor arc bound is obtained through a type I/II decomposition of τ_k, while the major arc contribution is evaluated by comparing the sum to a Dirichlet kernel and using an asymptotic for moments of the Dirichlet kernel. The additive-twist error in the major arcs is quoted from external results of Iwaniec–Kowalski and Matomäki–Radziwiłł–Tao.
Significance. If correct, the theorem gives a sharp, explicit asymptotic for all higher moments s>2 of the exponential sum with τ_k, with a positive leading constant and a power saving in X. This is a natural and useful result in the analytic theory of divisor functions, complementing earlier moment estimates for k-free numbers and k-th powers. The paper has clear structural strengths: the main term is computed from an explicit Dirichlet-kernel moment asymptotic, the constants γ_{ℓ,s,k} are defined by the major-arc expansion rather than fitted to data, and there are no free parameters tuned to force the claimed result. The minor arc argument is mostly self-contained and the reliance on standard type I/II bounds is reasonable. However, as written, the central claim is not fully established because the major-arc error term in Proposition 2.1 is asserted rather than derived, and the additive-twist estimate that controls the power saving is quoted from external references without a verification in this paper.
major comments (3)
- [§3, proof of Proposition 2.1, displayed equation after the first equality] The error term O(X^{s-1}P^{9/2+1/(k+1)}X^{-2/(k+1)+ε}) is asserted without derivation. To justify it, one must bound the integral over all major arcs of ||M(a/q+β)|^s - |Q_{k,q}(log X)v(β)|^s| dβ using the error term in Corollary 3.2. The paper gives no such estimate. A direct application of the trivial inequality ||A+E|^s-|A|^s| ≤ C_s|E|(|A|^{s-1}+|E|^{s-1}) and the bounds in Corollary 3.2 does not immediately produce the displayed P-exponent; the claimed cancellation or integration estimate is not evident. Since this error term directly determines the power saving δ_{s,k} in Theorem 1.1, this missing step is load-bearing and must be supplied.
- [§3, Proposition 3.1] Proposition 3.1 is the only source for the additive-twist error term with q-exponent 1/2+k/(k+1), and its proof is deferred to [4, Prop. 4.2] and [1, Thm. 4.16] without a derivation of the displayed error term. Because the size of the final power saving δ_{s,k} is obtained by balancing exactly this q-exponent against the minor arc bound, Theorem 1.1 is conditional on an external estimate that is not verified in the manuscript. The authors should either include a proof of Proposition 3.1 (or a precise derivation from the cited theorems) or explicitly state in Theorem 1.1 that this estimate is imported as a hypothesis.
- [§4, Proposition 4.1, proof] The reduction of the type I and type II cases to Lemmas 13.7 and 13.8 of [1] is sketched, but the hypotheses of those lemmas are not checked in the cases arising from the decomposition of τ_k. In particular, the dyadic ranges for N_j and the coefficient bounds should be stated explicitly, since the final minor arc bound depends on these ranges. This is a minor gap compared with the major-arc issue, but it should be filled for completeness.
minor comments (5)
- [Abstract] The abstract appears to omit the exponent s in the integral; it should read |∑_{n≤X}τ_k(n)e(nα)|^s dα, matching Theorem 1.1.
- [§3, Corollary 3.2] The polynomial Q_{k,q}(log X) is introduced without an explicit definition; the relation between the main term in Proposition 3.1 and the function Q_{k,q} obtained by partial summation should be stated.
- [§3, proof of Proposition 2.1] The binomial expansion of Q_{k,q}(log X)^s is used for real s without explicitly recording the requirement that the ratio of lower-order coefficients to the leading coefficient is bounded by 1; this holds for large X because the leading coefficient is ≫1/q, but it should be stated.
- [§4, proof of Proposition 4.1] The notation /BD[1,2X] and /BD_{I_j} is undefined in the printed text; these appear to denote characteristic functions and should be typeset as such.
- [§2, proof of Proposition 2.2] The bound ∫_0^1 |M(α)|^2 dα ≪ X(log X)^{O(1)} is quoted by Parseval without comment; this is standard for τ_k, but a brief justification would improve readability.
Circularity Check
No significant circularity: the asymptotic is derived from an external additive-twist estimate and a self-contained major-arc expansion; no fitted parameter is renamed as a prediction.
full rationale
The paper's central claim is an asymptotic for the s-th moment of the exponential sum of the higher divisor function. The main term is produced by a circle-method decomposition into major and minor arcs. The major-arc contribution in Proposition 2.1 is obtained by substituting the additive-twist expansion of Corollary 3.2, whose proof is deferred to Proposition 4.2 of [4] and Theorem 4.16 of [1], and then expanding the resulting polynomial powers in powers of log X. These coefficients are defined by an explicit analytic expansion, not fitted to data, and no parameter is tuned to force the claimed main term. The minor-arc bound of Proposition 4.1 is derived from a standard type I/type II decomposition of the divisor function and cited lemmas from Iwaniec-Kowalski. The cited results are external to this paper, do not assume Theorem 1.1, and are not authored by the present author. The only self-citation in the reference list ([5]) is not used as load-bearing evidence in the proof. The paper's proof has a nontrivial gap in passing from Corollary 3.2 to the displayed major-arc error term, and the final power saving genuinely depends on the quoted additive-twist q-exponent; however, dependence on an external theorem and an omitted estimate is a correctness risk, not circularity. There is no self-definitional step, no fitted input called a prediction, and no load-bearing self-citation chain.
Assumptions & free parameters
assumptions (5)
- standard math Theorem 4.16 in Iwaniec-Kowalski [1] provides the required estimate for sums of τ_k(n)e(an/q) with the stated main term and error term.
- standard math Proposition 4.2 in Matomäki-Radziwiłł-Tao [4] supplies the method and error term for the major arc expansion.
- standard math Lemmas 13.7 and 13.8 in Iwaniec-Kowalski give the type I and type II exponential sum bounds used in Proposition 4.1.
- domain assumption The error term from Voronoi summation for τ_k is as uniform in q as stated in Proposition 3.1.
- domain assumption The infinite series for γ_{s,k,ℓ} converges and its tail beyond P=X^η contributes only to the error term.
Cite this review
Pith. "Pith review of Moment estimates for the exponential sum with higher divisor functions." pith.science (2026). https://pith.science/paper/L3XNS6C2
@misc{pith2026190804286,
author = {Pith},
title = {Pith review of: Moment estimates for the exponential sum with higher divisor functions},
year = {2026},
howpublished = {\url{https://pith.science/paper/L3XNS6C2}},
note = {Machine review of arXiv:1908.04286}
}
abstract
We obtain asymptotic for the quantity $\int_0^1 \bigg|\sum_{n\le X}\tau_k(n)e(n\alpha)\bigg|d\alpha$ where $\tau_k(n) = \sum_{d_1\dots d_k = n} 1$. This follows from a quick application of the circle method. Along the way, we find minor arc bounds for the exponential sum with $\tau_k$, and asymptotics for high moments of the Dirichlet kernel.
Reference graph
Works this paper leans on
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[1]
H. Iwaniec and E. Kowalski, Analytic number theory , Amer. Math. Soc. Colloquium Publ. 53, Amer. Math. Soc., Providence RI, 2004
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[2]
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E. Keil. Moment estimates for exponential sums over k-fr ee numbers. Int. J. Number Theory. 9 (2013), 607619
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[4]
Correlations of the von Mangoldt and higher divisor functions I. Long shift ranges
K. Matom ¨ aki, M. Radziwiłł, and T. Tao. Correlations of the von Mangoldt and higher divisor func- tions I. Long shift ranges. arXiv:1707.01315, 2017
work page Pith review arXiv 2017
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[5]
M. Pandey . On the mean value of the magnitude of an exponen tial sum involving the divisor func- tion. arXiv:1901.07280, 2019
work page Pith review arXiv 1901
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[6]
R. C. Vaughan, T. D. W ooley . On the Distribution of Genera ting Functions. Bull. London Math. Soc. 30 (1998), 113122. DEPARTMENT OF MATHEMATICS , CALIFORNIA INSTITUTE OF TECHNOLOGY , PASADENA , CA 91125 E-mail address: mpandey@caltech.edu
work page 1998
Reviewed August 14, 2026 · model on record in the stance chip above.
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