REVIEW 2 major objections 5 minor 12 references
Note on a sum involving the divisor function
T0 review · 2 major / 5 minor · reviewed 2026-08-16 · deepseek-v4-flash
Pith's one-line read The paper proves an asymptotic formula for $\sum_{n \leq x^{1/c}} d(\lfloor x/n^c \rfloor)$ with error exponent $(2c+2)/(2c^2+5c+2)$ for $c<2/3$ and $5/(5c+6)$ for $c \geq 2/3$, improving the earlier bound for all $c>2/9$.
desk verdict Correct, modest improvement over Feng for c>2/9; the skipped verification in (3.7) actually checks out, so the paper deserves peer review. 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 mechanism is an exponential-sum lemma (Lemma 2.2) applied to the phase $f(z) = h x^{1/c}/(z+\delta)^{1/c}$ on each dyadic block $D<k \leq 2D$, together with an approximation lemma (Lemma 2.1) that replaces the sawtooth function $\psi(t) = t - \lfloor t \rfloor - 1/2$ by a short exponential sum with an explicit error. For the phase, the relevant size parameter is $F = h x^{1/c} D^{-1/c}$, and the lemma is used with $g=1$ to estimate $\sum_{D<k \leq 2D} d(k) e(h x^{1/c}/(k+\delta)^{1/c})$. The proof then writes the tail contribution $R_\delta(x)$ as a maximum over dyadic blocks and optimizes the free parameters $N$ and $H$, splitting the original sum at $N$ and truncating the sawtooth expansion at $H$, to balance the three error terms.
What would settle it
Verify condition (3.7) numerically for the chosen $H$ and $N$ across the full interval $D \in (N, x/N^c]$ for several values of $c$ (for example $c=1/2$ and $c=1$) at large $x$; any violation of $D^{3/4+1/c} x^{-1/c} \leq 1 \leq H \leq D^{3/2+1/c} x^{-1/c}$ would show that the application of Lemma 2.2 is not uniform, and the claimed exponent would not be established by this proof.
Extended reading notes
Core claim
The central discovery is that the error term in the asymptotic expansion of $S_{d,c}(x)$ can be pushed below the earlier exponent for all $c>2/9$ by choosing the splitting point $N$ and the exponential-sum cutoff $H$ more carefully. Theorem 1.1 states that $S_{d,c}(x) = x^{1/c} d_c + O_{\varepsilon,c}(x^{\theta_c + \varepsilon})$, where $\theta_c = (2c+2)/(2c^2+5c+2)$ for $0<c<2/3$ and $\theta_c = 5/(5c+6)$ for $c \geq 2/3$. The main term is $x^{1/c} d_c$, with $d_c = \sum_{k \geq 1} d(k)(k^{-1/c} - (k+1)^{-1/c})$, a constant independent of $x$. The argument splits the sum at a parameter $N$, rewrites the tail with the fractional-part identity, and bounds the resulting exponential sums uniformly over dyadic intervals, optimizing $H$ and $N$ to balance the error contributions.
Load-bearing premise
The proof rests on the uniform applicability of Lemma 2.2 to the phase $h x^{1/c}/(z+\delta)^{1/c}$ for every dyadic block $D \in (N, x/N^c]$ and every $h \leq H$, including the required asymptotic shape and the size range $D^{3/4} \ll F \ll D^{3/2}$; if the implied constants or the inequalities in condition (3.7) are not uniform in $c$ and $D$, the stated exponent does not follow.
Editorial extensions
If this is right
- For every real $c>2/9$, the asymptotic formula for $S_{d,c}(x)$ carries a strictly smaller error exponent than the previously available one, so the main term $d_c x^{1/c}$ is established with a higher power of $x$ in the error.
- At $c=1$, the theorem recovers the best known error term $O(x^{5/11+\varepsilon})$ for the sum $\sum_{n \leq x} d(\lfloor x/n \rfloor)$, making the result a direct generalization of that case.
- The main-term constant $d_c$ remains the same as in the earlier asymptotic, with the improvement confined entirely to the error term, which supports the expectation that $d_c x^{1/c}$ is the true leading behavior for all $c>0$.
- Because the exponent switches at $c=2/3$, the method yields two different optimal balances depending on whether the parameter $c$ is small or large.
Reading between the lines
- The same splitting-and-exponential-sum strategy could plausibly sharpen error terms for sums of the form $\sum_{n \leq x^{1/c}} f(\lfloor x/n^c \rfloor)$ for any arithmetic function $f$ with divisor-like growth whose associated exponential sums obey a similar estimate, such as generalized divisor functions or Fourier coefficients of modular forms.
- The optimized choices of $N$ and $H$ suggest that the true error exponent for this problem may be lower than $\theta_c$; a finer stationary-phase analysis of the same exponential sums, not attempted here, might produce a further improvement.
- A numerical scan of the admissible range for $H$ across all dyadic blocks $D \in (N, x/N^c]$ would reveal how close the present choice is to the limit of Lemma 2.2, and thus whether the bottleneck is the lemma or the optimization.
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper studies the sum S_{d,c}(x) = \sum_{n\le x^{1/c}} d(\lfloor x/n^c\rfloor), where d(n) is the divisor function and c>0. The main result, Theorem 1.1, claims an asymptotic formula S_{d,c}(x) = d_c x^{1/c} + O_{\varepsilon,c}(x^{\theta_c+\varepsilon}) with d_c = \sum_{k\ge1} d(k)(k^{-1/c} - (k+1)^{-1/c}) and exponent \theta_c = (2c+2)/(2c^2+5c+2) for 0<c<2/3, \theta_c = 5/(5c+6) for c\ge2/3. The proof splits the range of n at N, converts the sum to one over k = \lfloor x/n^c\rfloor, applies Vaaler's approximation to the fractional part and Jutila's exponential-sum estimate with the divisor function, and then optimizes the parameters N and H. The abstract and theorem also state that this improves Feng's error term for c>2/9 and, at c=1, recovers Stucky's exponent 5/11.
Significance. If the proof is correct, the result is a genuine improvement in an active line of work on sums of arithmetic functions over \lfloor x/n\rfloor-type arguments. The key elements are standard tools (Vaaler's approximation, Jutila's exponential-sum lemma) and a careful parameter optimization; the paper is concise and the claimed exponent is an explicit, falsifiable quantity. The manuscript also claims, and it is straightforward to verify, that the new exponent is uniformly better than Feng's for all c>2/9. The proof strategy is transparent, and there are no fitted constants or ad hoc hypotheses beyond the standard use of cited lemmas. However, as discussed in the major comments, the proof as written contains a sign error in an intermediate identity and a load-bearing applicability condition that is asserted but not verified.
major comments (2)
- [Section 3, Eq. (3.3)] The displayed identity in (3.3) has a sign error. Using \psi(t)=t-\lfloor t\rfloor-1/2, one has \lfloor x^{1/c}/k^{1/c}\rfloor - \lfloor x^{1/c}/(k+1)^{1/c}\rfloor = x^{1/c}(k^{-1/c}-(k+1)^{-1/c}) - \psi(x^{1/c}/k^{1/c}) + \psi(x^{1/c}/(k+1)^{1/c}). Hence the remainder terms should appear as -R_0(x)+R_1(x), not +R_0(x)-R_1(x). The subsequent argument bounds only |R_0| and |R_1|, so the final estimate is unaffected, but the identity must be corrected.
- [Section 3, condition (3.7)] The proof of the exponential-sum bound (3.6) depends on the claim that the chosen parameters H and N satisfy (3.7) for every D in (N,x/N^c]. This is asserted without proof. The condition is essential: it is exactly the range requirement of Lemma 2.2 that guarantees the phase f(z)=h x^{1/c}(z+\delta)^{-1/c} has the required growth and asymptotic form. Please supply the verification. For 0<c<2/3, one should check that the lower bound at the upper endpoint D=x/N^c gives an exponent (3c+4)(3c+2)/(4c(2c^2+5c+2))-1/c = (c-2)/(4(2c^2+5c+2)) < 0, and that the upper bound at D=N gives H = N^{3/2+1/c}/x^{1/c} exactly; for c\ge2/3, the analogous endpoint computations give an exponent difference (8-12c)/(8c(5c+6)) \le 0. Additionally, for \delta=1 the phase must be written as B z^{-1/c}(1+O(F^{-1/3})) with B=h x^{1/c}; this requires noting that the relative error O(1/D) is O(F^{-1/3}) under the upper bound H \ll D^{3/2+1/c}/x^{1/c}. Without these checks, the applicability of Lemma 2.2, and hence the claimed error exponent, is not established in the manuscript.
minor comments (5)
- [Section 1, Notation] The definition of \psi(t) reads '\psi(t) := x - [x] - 1/2'; this should be '\psi(t) := t - [t] - 1/2'.
- [Section 3, Eq. (3.4)] In the definition of S_{\delta,c}(x,D), the argument of \psi contains an extraneous factor h; it should be \psi(x^{1/c}/(k+\delta)^{1/c}). The parameter h first appears properly in (3.5) when Vaaler's approximation is applied.
- [Section 3, parameter choice] The expressions for H and N are real numbers, while Lemma 2.1 and the sums in (3.5) require H to be an integer. One should take H, N to be integers of comparable size (e.g., floor values) and note that the error estimates are unchanged.
- [Abstract and Theorem 1.1] The abstract writes the exponent as \max\{(2c+2)/(2c^2+5c+2),5/(5c+6)\}, while the theorem states the same exponent in piecewise form. The two agree, but stating the formula consistently in both places would avoid confusion.
- [Section 2, Lemma 2.1 citation] Lemma 2.1 is attributed to Vaaler via '[5, Theorem A.6]', but reference [5] is the Graham-Kolesnik monograph, not Vaaler's original paper. The citation is acceptable, but a direct reference to Vaaler's theorem would be cleaner.
Circularity Check
No circularity: the main term is an explicit convergent series and the error exponent is optimized from two external exponential-sum lemmas; the unverified range condition (3.7) is a gap, not a circular reduction.
full rationale
The paper's main term is obtained from an exact identity rewriting the sum over n as a sum over k, producing the constant d_c = sum_{k>=1} d(k)(k^{-1/c} - (k+1)^{-1/c}) plus remainder terms. This constant is an absolutely convergent series defined directly from divisor values; it is not fitted, predicted, or imposed. The error exponent in Theorem 1.1 is obtained by estimating the remainder terms R_delta(x) with two external tools: Vaaler's approximation of the sawtooth function (Lemma 2.1, cited to Graham-Kolesnik) and Jutila's exponential-sum lemma (Lemma 2.2, cited to Jutila's published Tata Institute lectures). Neither lemma is a self-citation, and neither lemma is invoked as a renamed version of the target theorem. The only same-author citation in the introduction, Wu [11], supplies an earlier general bound for S_f(x) and is motivational and contextual, not load-bearing in the proof of Theorem 1.1. The proof then balances two error terms by choosing the split parameter N and the Vaaler truncation parameter H; this is an optimization over parameters, not a fit of the claimed error exponent to observed data. The main gap in the paper is the assertion that condition (3.7) holds for the chosen N and H, with the text merely saying 'it is easy to see' and not displaying the endpoint inequalities. That is a serious correctness risk, because (3.7) controls whether Jutila's lemma is applicable at every dyadic D in the required range and for delta = 0 and delta = 1. But a missing verification is not circularity: if the condition fails, the proof does not establish the theorem, whereas true circularity would make the theorem true by definition or by self-citation. Since no equation in the derivation reduces to its own input, no fitted parameter is renamed as a prediction, and no load-bearing claim rests on the author's own prior work, the paper exhibits no significant circularity.
Assumptions & free parameters
assumptions (3)
- standard math Vaaler's approximation for the fractional part function (Lemma 2.1)
- standard math Jutila's exponential sum estimate (Lemma 2.2)
- standard math Bound for the divisor function d(n) << n^epsilon for any epsilon > 0
Cite this review
Pith. "Pith review of Note on a sum involving the divisor function." pith.science (2026). https://pith.science/paper/4S5IFLFM
@misc{pith2026250501645,
author = {Pith},
title = {Pith review of: Note on a sum involving the divisor function},
year = {2026},
howpublished = {\url{https://pith.science/paper/4S5IFLFM}},
note = {Machine review of arXiv:2505.01645}
}
abstract
Let $d(n)$ be the divisor function and denote by $[t]$ the integral part of the real number $t$. In this paper, we prove that $$\sum_{n\leq x^{1/c}}d\left(\left[\frac{x}{n^c}\right]\right)=d_cx^{1/c}+\mathcal{O}_{\varepsilon,c} \left(x^{\max\{(2c+2)/(2c^2+5c+2),5/(5c+6)\}+\varepsilon}\right),$$ where $d_c=\sum_{k\geq1}d(k)\left(\frac{1}{k^{1/c}}-\frac{1}{(k+1)^{1/c}}\right)$ is a constant. This result constitutes an improvement upon that of Feng.
Reference graph
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Reviewed August 16, 2026 · model on record in the stance chip above.
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