{"id":"c5cdbff8-1e8e-4452-97ab-16711b8de0dc","arxiv_id":"2505.01645","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For a sum of the divisor function over rounded quotients x/n^c, this note proves an asymptotic formula with an error exponent that improves Feng's estimate for all c > 2/9.","lead":"A short number theory note proves a sharper error term for an asymptotic formula that counts divisors of the rounded values x/n^c. The improvement over Feng's estimate holds for all exponents c > 2/9 and recovers a recent c = 1 result.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The central estimate hinges on the unverified range condition (3.7) for Jutila's Lemma 2.2; absent a constant-level check at D=N and for delta=1, the claimed error exponent is not established.","rationale":"After checking the parameter balance algebraically, the exponents in the three error terms do balance as stated, and the comparison with Feng's exponents is plausible for c>2/9. The weakest point is not the balance but the unproved applicability of Lemma 2.2: condition (3.7) is asserted without verification, and both the delta=1 expansion and the lower/upper F-range involve constants that are not tracked. This is exactly the reader's weakest assumption. The concern is non-fatal in the sense that explicit calculation strongly suggests the inequalities hold with room at the endpoints, but the manuscript as written asks the reader to take the key technical condition on faith. Other weaknesses (abstract overclaim for c<=2/9, typographical errors in psi(t) and in the definition of S_delta) do not affect the central estimate. Therefore the conditional verdict survives unchanged, pending the requested verification.","tokens_in":5041,"tokens_out":39744,"duration_ms":366608,"concrete_test":"Symbolically substitute N=x^{2(1+c)/(2c^2+5c+2)} and H=x^{3/4-1/(4c)}N^{1/(4c)-3c/4-3/8} into (3.7) for 0<c<2/3, and N=x^{5/(5c+6)} and H=x^{3/8}N^{-3c/8-1/4} for c>=2/3. Check at the endpoints D=N and D=x/N^c: verify (a) D^{3/4+1/c} <= c1 x^{1/c}, (b) 1 <= H <= D, (c) H <= c2 D^{3/2+1/c}/x^{1/c}, and (d) for delta=1, 1/D <= c3 F^{-1/3} for h=1 and h=H, with explicit constants c1, c2, c3. If any inequality fails for a representative c (e.g., c=0.01, 0.6, 2/3, 1, 10), then (3.6) is not available and the error exponent must be re-derived.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The proof of Theorem 1.1 reduces R_delta to the exponential-sum bound (3.6), which is only valid if Lemma 2.2 is applicable for every dyadic D in (N, x/N^c] and every h <= H. Concretely, the phase f(z)=h x^{1/c}(z+delta)^{-1/c} must satisfy f(z)=B z^{-1/c}(1+O(F^{-1/3})) with F=h x^{1/c}D^{-1/c} and D^{3/4} << F << D^{3/2}. The paper states this as condition (3.7) and says 'it is easy to see' the chosen N and H verify it, but no inequality is shown. Three specific points need checking: (i) for delta=1, the shift by 1 produces a relative error O(1/D) that is only O(F^{-1/3}) if F << D^3 with a controlled constant; (ii) the h=1 lower bound requires D^{3/4+1/c} << x^{1/c} uniformly, with the maximum at D=x/N^c; (iii) H must simultaneously satisfy 1 <= H <= D for every D and H << D^{3/2+1/c}/x^{1/c}, the latter being tight at D=N. The paper gives no constants or endpoint verification, so the balanced exponents in (3.11) and (3.12) rest on an unproved applicability condition.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":5336,"tokens_out":22043,"duration_ms":182613,"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":[{"comment":"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":"Section 3, Eq. (3.3)"},{"comment":"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.","section":"Section 3, condition (3.7)"}],"minor_comments":[{"comment":"The definition of \\psi(t) reads '\\psi(t) := x - [x] - 1/2'; this should be '\\psi(t) := t - [t] - 1/2'.","section":"Section 1, Notation"},{"comment":"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":"Section 3, Eq. (3.4)"},{"comment":"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.","section":"Section 3, parameter choice"},{"comment":"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":"Abstract and Theorem 1.1"},{"comment":"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.","section":"Section 2, Lemma 2.1 citation"}],"recommendation":"major_revision","confidential_remarks":"I have independently checked the exponent balance and the endpoint verification of condition (3.7); the claimed theorem appears to be correct. The main required revision is to supply the missing verification of (3.7) (including the \\delta=1 phase expansion) and to correct the sign error in (3.3). Both issues are local and well within the scope of the manuscript, so I am confident the paper can be made acceptable."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper is a correct, modest note. It proves a new error exponent for S_{d,c}(x)=∑_{n≤x^{1/c}} d([x/n^c]): θ_c=(2c+2)/(2c^2+5c+2) for 0<c<2/3, and 5/(5c+6) for c≥2/3. This improves Feng's exponent for all c>2/9 and recovers Stucky's 5/11 at c=1. The technique is not new—Vaaler's approximation plus Jutila's exponential-sum lemma—but the parameter balance is clean and yields a genuine improvement over Feng.\n\nI checked the one substantive concern: the applicability of Jutila's lemma, condition (3.7). The stress-test worries that the inequalities are unverified. Plugging in the chosen H and N, both endpoints work. At D=x/N^c, the lower bound D^{3/4+1/c} ≪ x^{1/c} holds with room; at D=N, the upper bound H x^{1/c} ≪ D^{3/2+1/c} is tight but exact. The δ=1 shift changes the phase only by O(1/D), which is absorbed in the O(F^{-1/3}) error allowed by the lemma. So the central estimate stands; the 'easy to see' is true but the author should show the arithmetic.\n\nThe weak spots are presentation-level. The abstract says 'improvement upon Feng' without the c>2/9 qualifier, which is false for c≤2/9. There is a sign error in (3.3): the R_0 and R_1 should be swapped, but the two are bounded identically and the conclusion is unaffected. The definition of ψ(t) writes x instead of t. These are minor typos.\n\nThe result is honest progress in a narrow branch. The citation pattern is ordinary, and the comparison to Feng and Stucky is clearly drawn. It is not a paper for a general reading group, but a specialist in exponential sums would find it a useful and correct data point.\n\nRecommendation: send it to a competent referee rather than desk-rejecting. Once the typos and the abstract are fixed, it should be accepted.","headline":"Correct, modest improvement over Feng for c>2/9; the skipped verification in (3.7) actually checks out, so the paper deserves peer review.","tokens_in":5880,"tokens_out":13731,"would_cite":true,"duration_ms":115367,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11N37","11L07"],"pacs":[],"model":"deepseek-v4-flash","headline":"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 $c>2/9$.","keywords":["divisor function","asymptotic formula","exponential sum","integral part","error term","floor function","summatory function"],"falsifier":"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.","tokens_in":4815,"feed_emoji":"🔢","tokens_out":9961,"duration_ms":86855,"temperature":0.7,"pith_summary":"This note studies the summatory function $S_{d,c}(x) = \\sum_{n \\leq x^{1/c}} d(\\lfloor x/n^c \\rfloor)$, where $d$ is the divisor function and $c$ is any positive real number. The paper proves an asymptotic formula $S_{d,c}(x) = x^{1/c} d_c + O_{\\varepsilon,c}(x^{\\theta_c + \\varepsilon})$, with $d_c = \\sum_{k \\geq 1} d(k)(k^{-1/c} - (k+1)^{-1/c})$ a convergent constant and $\\theta_c$ equal to $(2c+2)/(2c^2+5c+2)$ for $0<c<2/3$ and $5/(5c+6)$ for $c \\geq 2/3$. This improves the previously known error exponent for every $c>2/9$, and at $c=1$ it reproduces the sharpest known exponent $5/11$ for the ordinary divisor sum $S_d(x)$. The interest is that the improvement is achieved unconditionally from standard exponential-sum estimates, with no new hypotheses.","feed_headline":"New error exponent improves divisor-sum estimate for all c > 2/9","feed_subtitle":"A sharper asymptotic for summing the divisor function over floor(x/n^c), beating the previous bound whenever c > 2/9.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"The earlier asymptotic formula for $S_{d,c}(x)$ whose error exponent this paper improves; it supplies the benchmark to beat.","marker":"[4]"},{"why":"Contains the approximation lemma (Lemma 2.1) used to express the fractional-part function $\\psi$ as a short exponential sum with an explicit error.","marker":"[5]"},{"why":"Supplies the exponential-sum estimate (Lemma 2.2) that controls the weighted sums of $d(k)$ over dyadic intervals.","marker":"[6]"},{"why":"The $c=1$ divisor-sum case whose error exponent $5/11$ the present result generalizes and matches at $c=1$.","marker":"[10]"}],"fun_headline_variants":["New error exponent beats previous divisor-sum bound for c > 2/9","Sharper divisor-sum error term improves on Feng for c > 2/9","Divisor-sum exponent improved for all c > 2/9, beating prior result","Error exponent cut in divisor sum for all c > 2/9","Improved divisor-sum estimate for c > 2/9 sharper than Feng's"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["New error exponent beats previous divisor-sum bound for c > 2/9","Sharper divisor-sum error term improves on Feng for c > 2/9","Divisor-sum exponent improved for all c > 2/9, beating prior result","Error exponent cut in divisor sum for all c > 2/9","Improved divisor-sum estimate for c > 2/9 sharper than Feng's"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001097,"raw_usage":{"total_tokens":4566,"prompt_tokens":918,"completion_tokens":3648,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":534,"completion_tokens_details":{"reasoning_tokens":3545}},"tokens_in":534,"tokens_out":3648,"duration_ms":24132,"temperature":1.0,"reasoning_tokens":3545,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T04:15:02.236623+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"The earlier asymptotic formula for $S_{d,c}(x)$ whose error exponent this paper improves; it supplies the benchmark to beat."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Contains the approximation lemma (Lemma 2.1) used to express the fractional-part function $\\psi$ as a short exponential sum with an explicit error."},{"cited_title":"Jutila, Lectures on a Method in the Theory of Exponential Sums , Tata Institute of Fundamental Research Lectures on Mathematics and Physics, vol","cited_arxiv_id":null,"evidence_quote":"Supplies the exponential-sum estimate (Lemma 2.2) that controls the weighted sums of $d(k)$ over dyadic intervals."},{"cited_title":"Stucky, The fractional sum of small arithmetic functions, J","cited_arxiv_id":null,"evidence_quote":"The $c=1$ divisor-sum case whose error exponent $5/11$ the present result generalizes and matches at $c=1$."}],"review_version":1}