{"id":"4eeacfb8-6027-4f21-8efb-837efba6a46e","arxiv_id":"1908.05598","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For the divisor function with congruence conditions, the error term changes sign in every interval of length O(√T) and has many intervals of length √T log^{-7} T where it exceeds c t^{1/4}.","lead":"The paper proves that the error term in a congruence-restricted divisor problem changes sign in every interval of length a constant times √T, even when a small perturbation is added. It also shows long stretches in [T,2T] where this error term stays above c t^(1/4) or below -c t^(1/4) occur frequently, extending classical results.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The sign-change theorem rests on an unproved Voronoi-type formula imported from the authors' earlier paper; Theorem 2.1's proof is also deferred, so the central claim is conditional.","rationale":"The reader's conditional verdict is appropriate, and the weakest point is indeed Lemma 3.1. All theorems either use it directly (Theorem 2.2 via Lemma 4.1) or use Theorem 2.1, whose proof is deferred. The final omitted step in Section 4 is a real but minor completeness gap; it does not change the load-bearing status because the missing sign-change inference from a smoothed local average is standard in the style of Heath-Brown and Tsang. A careful independent verification of Lemma 3.1, including the uniformity in q1,q2, would settle whether the paper's foundation is sound. I do not see a reason to move beyond conditional based on this text.","tokens_in":14658,"tokens_out":22110,"duration_ms":215021,"concrete_test":"Independently re-derive Lemma 3.1 from the proof in [9] and verify that the stated error O(log^3(q1q2T)) is uniform for x in [T/2,T], for all 1<=ri<=qi, and for the allowed H,y ranges. As a first check, specialize to q1=q2=1, r1=r2=1 and confirm that (3.1) reduces to the classical truncated Voronoi formula for Delta(x) with matching constants and cutoffs; then test the uniformity by repeating the derivation with large q1,q2. If the reduction or the uniformity fails, the foundation of Theorem 2.2 collapses.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is Theorem 2.2, and its proof in Section 4 is a smoothed-local-average identity, Lemma 4.1. That lemma is obtained by substituting the Voronoi-type expansion of Lemma 3.1 into the definition of Delta**(t). Lemma 3.1 is not proved in this manuscript; it is quoted from the authors' previous paper [9], including the O(log^3(q1q2T)) error term and the sums G12, G21. The paper also does not prove Theorem 2.1: Section 3 states Lemma 3.1 and then only says the moment asymptotic follows with the approach of Liu [11]. Corollary 2.1 is subsequently used in Lemmas 6.1 and 6.2, so Theorems 2.3 and 2.4 inherit the same unverified input. If Lemma 3.1, or its uniform error and G-estimates, is incorrect, the main term in Lemma 4.1 can be destroyed and Theorem 2.2 does not follow. A secondary gap is that Section 4 ends immediately after Lemma 4.1; the final sign-change argument is not written out, although it is probably routine once Lemma 4.1 is granted. The load-bearing uncertainty is therefore the imported Voronoi formula.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the error term Δ(x; r1,q1,r2,q2) in the summatory function of the divisor function d(n; r1,q1,r2,q2) with congruences. The main results are Theorem 2.1, giving power moment asymptotics for Δ(q1q2x; r1,q1,r2,q2) for integer k with 3 ≤ k ≤ 9; Theorem 2.2, asserting that Δ(q1q2t; r1,q1,r2,q2)+f(t) changes sign in every interval [T, T+c2√T] for small perturbations f(t) with |f(t)| ≤ c1 t^{1/4}; Theorem 2.3, asserting many subintervals of length c4√T log^{-7}T in [T,2T] on which ±Δ > c5 t^{1/4}; and Theorem 2.4, an Ω-type lower bound for the deviation F_k of the k-th moment from its asymptotic. The proofs are built on a Voronoi-type expansion (Lemma 3.1) quoted from the authors' earlier paper [9], a smoothed local-average identity (Lemma 4.1), and a short-interval mean-square estimate (Lemma 5.2).","tokens_in":14901,"tokens_out":2637,"duration_ms":26144,"significance":"If the results are correct, the paper provides a natural congruence-conditioned analogue of the Heath–Brown–Tsang sign-change theorem for the classical divisor problem, with the same order of interval length √T and, in Theorem 2.3, the same logarithmic loss log^{-7}T. The power-moment asymptotic for k = 3,...,9 is stated with explicit constants C_k and the short-interval variance estimate is developed in a way that may be useful for further work. The paper makes its dependence on the authors' prior work explicit, and no circularity or parameter-fitting is apparent. The main strength is that the analytic skeleton from the classical case is carried over to the congruence-conditioned setting, but the central input remains the unproved Voronoi-type formula from [9].","major_comments":[{"comment":"Lemma 3.1 is the Voronoi-type expansion (3.1) for Δ(q1q2x; r1,q1,r2,q2), quoted from [9] without proof. This lemma is load-bearing for every subsequent theorem: it is used to prove Theorem 2.1 and Corollary 2.1, to derive Lemma 4.1 in §4, and indirectly through Corollary 2.1 in Lemmas 6.1 and 6.2. The uniform error term O(log^3(q1q2T)) and the estimates for G12 and G21 are essential to the later arguments. The manuscript should either include a proof of Lemma 3.1 or clearly state it as a standing hypothesis; in its current form the central claims are conditional on an external result whose proof is not reproduced or summarized.","section":"§3, Lemma 3.1"},{"comment":"Theorem 2.1 is not proved in the manuscript: the text says only that it follows from Lemma 3.1 'with the approach of Liu [11]'. Corollary 2.1 is a direct consequence and is used later in Lemma 6.1 and Lemma 6.2, and in the proof of Theorem 2.3 via the fourth moment. Since the moments of Δ are a main result and feed into the sign-change and Ω-results, the proof should at least sketch the key steps, including the treatment of the cross terms in the expansion for Δ^k and the source of the explicit constants C_k. Without this, the validity of Theorems 2.3 and 2.4 is also unresolved.","section":"§3, proof of Theorem 2.1"},{"comment":"Section 4 proves Lemma 4.1 and then stops; the actual sign-change argument for Theorem 2.2 is not written out. Lemma 4.1 gives the smoothed local average of Δ**(t), but the theorem requires the existence of points t1, t2 in [T,T+c2√T] with Δ values of opposite signs of size ≥ c1 t^{1/4}. This final step is presumably routine from Lemma 4.1, but it is not present. The proof of Theorem 2.2 is therefore incomplete as written.","section":"§4, proof of Theorem 2.2"}],"minor_comments":[{"comment":"There are several typos in Section 5: 'Therefor' in the paragraph after (5.4), 'Lamma 5.1' before equation (5.9), and 'we we can deduce' in Section 6 before Lemma 6.2.","section":"§5, Lemma 5.2"},{"comment":"In the definition of F_k, the integral is written as ∫_T^1 Δ^k(...)dx, which should presumably be ∫_1^T to match the preceding notation; please correct the limits.","section":"§7"},{"comment":"The proof of Lemma 6.2 is compressed; in particular the inequality 'Δ*(q1q2u2) - Δ*(q1q2u1) ≥ -O((u2-u1) log T)' needs a justification or a reference, and the splitting argument with λ and b is only sketched.","section":"§6, proof of Lemma 6.2"},{"comment":"Theorem 2.1 contains a grammatical error ('If A0 > 9 satisfies ... then ...') and the condition 'T ≫ (q1q2)ε is large enough' should be stated more precisely, for example specifying the ε in the exponent.","section":"§2, Theorem 2.1"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is a sequel to the authors' paper [9], and the referee's main concern is the degree to which the present paper depends on unproved results from that earlier work. It would strengthen the paper considerably if the authors provided a self-contained statement of the Voronoi-type expansion with all uniformity conditions, or at least a detailed proof of the new key steps. I recommend major revision rather than rejection because the framework is coherent and the missing parts appear to be fillable, but the current text does not yet constitute a complete proof of the stated theorems."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know about 1908.05598. First, it genuinely breaks new ground: no one had written down sign-change and many-large-value theorems for Δ(x;r1,q1,r2,q2), and the perturbation version in Theorem 2.2 is a nice touch. Second, the paper is not self-contained at the load-bearing point. The central input is Lemma 3.1, a Voronoi-type expansion for Δ(q1q2x;…), quoted from the authors' earlier paper [9] without proof. If that lemma or its uniform error estimates are wrong, everything downstream—Theorem 2.1, Lemma 4.1, and the sign-change theorem—fails. The reader's report is right to put the burden there.\n\nWhat the paper does well: the adaptation of Heath-Brown–Tsang and Tsang–Zhai to the two-congruence setting is careful, and the computations in Section 5 (the short-interval integral) are reasonably detailed. The Hilbert-inequality argument follows a known template without obvious missteps. The constants are explicit and no free parameters are fitted to make the theorems work; the claims are concrete and falsifiable in principle.\n\nSoft spots, in order of size. (1) Lemma 3.1 is quoted, not proved. Since this is the authors' own prior result, it's not circular, but the paper would be stronger if it either restated the proof or at least stated the lemma in a form that a referee can check against [9]. (2) Theorem 2.1's proof is essentially '... with the approach of Liu [11]'—that's a sketch, not a proof. Corollary 2.1 follows from it and is then used in Lemmas 6.1 and 6.2, so the later theorems inherit the gap. (3) Section 4 ends right after Lemma 4.1; the final step from the smoothed local average to the sign-change conclusion is not written out. It probably follows the Heath-Brown–Tsang argument almost verbatim, but a referee will need to fill it in. (4) Minor: Lemma 6.1 invokes 'Corollary 2.1 with k=2,4', but the corollary only covers 3 ≤ k ≤ 9. The k=2 case is covered by earlier work (e.g., Müller–Nowak or the authors' own [9]), so this is a misstatement, not a fatal error. Also a handful of typos.\n\nIs the central argument sound? I think likely yes. The technique is mature, the new input is a careful bookkeeping of the congruence phases, and the error terms are estimated in the standard way. But because the key lemma is unproved here, I would not accept the paper as-is. It deserves a serious referee, and the referee should demand that Lemma 3.1 be proved or at least quoted with enough of [9]'s argument to make its validity transparent.\n\nWho this is for: analytic number theorists working on divisor problems and fine-grained behavior of error terms. Not a breakthrough, but a solid contribution that extends known theorems to a natural variant.\n\nRecommendation: send it to peer review, conditional on the authors filling the gaps. If I were the editor, I'd ask for a revision before acceptance.","headline":"New sign-change and large-value theorems for the congruence-conditioned divisor error term, worth refereeing, but the central Voronoi-type lemma is imported from the authors' earlier work without proof and Theorem 2.1 is only sketched.","tokens_in":15462,"tokens_out":3919,"would_cite":false,"duration_ms":34704,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11N37","11P21"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that the error term in the divisor problem with congruence conditions changes sign in every interval of length C√T, even after adding a perturbation, and that it has large positive and negative excursions on many short…","keywords":["divisor problem","congruence conditions","error term","power moments","sign changes","Voronoi formula","short intervals"],"falsifier":"Take one concrete pair, say $q_1=2,q_2=3$, and compute $\\Delta(q_1q_2t;r_1,q_1,r_2,q_2)$ at many points in a large interval $[T,T+C\\sqrt{T}]$. If for some large $T$ there is no point where the function, even after adding a perturbation $|f|\\le c_1 t^{1/4}$, takes both signs, Theorem 2.2 is false. A more targeted check is to evaluate Lemma 3.1 numerically for moderate $T$ and compare its right-hand side to the directly computed error term: an error larger than $O(\\log^3(q_1q_2T))$ outside the claimed bounds would invalidate the foundation of all four theorems.","tokens_in":14447,"feed_emoji":"🔢","tokens_out":11088,"duration_ms":100531,"temperature":0.7,"pith_summary":"This paper counts factorizations n=n1n2 with n1≡r1 (mod q1) and n2≡r2 (mod q2), and studies the error term $\\Delta(x;r_1,q_1,r_2,q_2)$ between that count and its smooth main term. It proves three things about the rescaled error term $\\Delta(q_1q_2x;r_1,q_1,r_2,q_2)$: its k-th power moments are asymptotic to $C_k T^{1+k/4}$ with nonzero explicit constants for every integer 3≤k≤9; it changes sign inside every interval $[T,T+C\\sqrt{T}]$ for sufficiently large T even after adding any perturbation $|f(t)|\\le c_1 t^{1/4}$; and the sets where it exceeds $c_5 t^{1/4}$ or falls below $-c_5 t^{1/4}$ each contain many disjoint short intervals and have measure $\\gg T$ in $[T,2T]$. These results bring the congruence-condition divisor problem to the same level of oscillation theory as the classical divisor problem, where the same $\\sqrt{T}$ sign-change window and the same kind of long intervals of one sign are known. If the theorems are right, the error term fluctuates at amplitude at least a constant times $t^{1/4}$ in every interval of length about $\\sqrt{T}$, while also containing many shorter subintervals where it keeps a single sign.","feed_headline":"Divisor error term flips sign in every √T window","feed_subtitle":"Even after adding a small perturbation, the error term must exceed t^{1/4} and fall below −t^{1/4} inside each such window.","key_machinery":"The engine of the paper is a Voronoi-type formula, stated as Lemma 3.1 and quoted from the authors' earlier paper [9]. It writes $\\Delta(q_1q_2x;r_1,q_1,r_2,q_2)$ as $x^{1/4}$ times a main oscillatory sum over $n=hl$ of $\\cos(4\\pi\\sqrt{nx}-2\\pi(hr_2/q_2+lr_1/q_1+1/8))/n^{3/4}$, truncated at $n\\le y$, plus two short-tailed sums $R_{12},R_{21}$, plus two error terms $G_{12},G_{21}$ involving Diophantine closeness of $q_1x/n_1-r_2/q_2$ and $q_2x/n_2-r_1/q_1$. This expansion turns the error term into an almost periodic trigonometric sum whose largest term, $n=1$, carries the phase $4\\pi t-2\\pi(r_2/q_2+r_1/q_1+1/8)$. To extract its sign, the paper integrates $\\Delta$ against the kernel $K_\\zeta(u)=(1-|u|)(1+\\zeta\\sin(4\\pi\\alpha u))$ with $\\zeta=\\pm1$; the sine factor selects the $n=1$ term while the triangular weight kills the other frequencies at cost $O(\\alpha^{-2})$. The remaining machinery is quantitative: a second-moment bound for increments of the error term (Lemma 5.2, built with Hilbert's inequality and splitting arguments) controls how much the error term can change over short shifts, and Hölder and Cauchy-Schwarz comparisons of second and fourth moments force the positive and negative parts of the function to both have large measure.","core_discovery":"In the paper's own terms, the discovery is that the error term $\\Delta(q_1q_2x;r_1,q_1,r_2,q_2)$ is a full analogue of the classical divisor error term: it has the same power moments and the same oscillation scale. Theorem 2.1 gives, for every fixed integer 3≤k≤9, the asymptotic $\\int_1^T \\Delta^k(q_1q_2x;r_1,q_1,r_2,q_2)\\,dx = C_k T^{1+k/4}(1+o(1))$, with $C_k\\asymp 1$ explicit, assuming a high-moment upper bound of order $T^{1+A_0/4+\\varepsilon}$. Theorem 2.2 is the sharpest oscillation statement: for a sufficiently large constant $c_2$ and a sufficiently small $c_1$, any perturbation $|f(t)|\\le c_1 t^{1/4}$ still leaves $\\Delta(q_1q_2t;r_1,q_1,r_2,q_2)+f(t)$ changing sign at least once in every interval $[T,T+c_2\\sqrt{T}]$; in particular the unperturbed error term takes values $\\ge c_1 t^{1/4}$ and $\\le -c_1 t^{1/4}$ inside every such interval. Theorem 2.3 proves that in $[T,2T]$ there are at least $c_3\\sqrt{T}\\log^{-7}T$ disjoint subintervals of length $c_4\\sqrt{T}\\log^{-7}T$ on which $\\pm\\Delta>c_5 t^{1/4}$ throughout, so each sign occupies measure $\\gg T$. Theorem 2.4 converts this into an $\\Omega$-result: for every odd k≥2 there is a point $X\\in[T,2T]$ where the k-th-moment error term $F_k(q_1q_2X;r_1,q_1,r_2,q_2)$ is $\\gg X^{1/2+k/4}\\log^{-7}X$, even though the moment asymptotics are proven only for 3≤k≤9.","pith_inferences":["The mechanism used here is transportable: any error term admitting a Voronoi-type expansion and a short-interval second-moment bound should yield the same $\\sqrt{T}$ sign-change dichotomy. This is an extrapolation beyond the paper, not one of its claims.","The perturbation robustness in Theorem 2.2 suggests that the sign change is driven by the $n=1$ frequency in the Voronoi expansion. A testable extension would be to see whether a larger perturbation, say $|f(t)|\\le c_1 t^{1/4}\\log t$, still forces a sign change inside the same window, or whether the window length must grow.","The gap between the guaranteed sign-change window length $\\sqrt{T}$ and the no-sign-change intervals of length $\\sqrt{T}\\log^{-7}T$ suggests a critical scale of the form $\\sqrt{T}\\log^{-a}T$ for some exponent $a$; a numerical experiment for a fixed pair $(q_1,q_2)$ could estimate that exponent. This is an editorial inference."],"forward_implications":["For every fixed residue pair, the k-th moment of $\\Delta(q_1q_2x;r_1,q_1,r_2,q_2)$ is asymptotically $C_k T^{1+k/4}$ for 3≤k≤9, so the error term's distribution matches the classical divisor error term at the level of moments.","In every window $[T,T+C\\sqrt{T}]$, the error term cannot stay on one side of zero: it attains values above $c_1 t^{1/4}$ and below $-c_1 t^{1/4}$, and the same holds even after adding any perturbation of size at most $c_1 t^{1/4}$.","Both the positive and negative excursions are common: the set of $t\\in[T,2T]$ with $\\pm\\Delta>c_5 t^{1/4}$ has measure $\\gg T$, made of $\\gg\\sqrt{T}\\log^{-7}T$ disjoint intervals of length $\\asymp\\sqrt{T}\\log^{-7}T$.","The error term in the odd power-moment asymptotic formula is $\\gg X^{1/2+k/4}\\log^{-7}X$ for every odd k≥2, so the moment expansion cannot be too accurate at any single point $X$ in $[T,2T]$.","The sign-change threshold at scale $\\sqrt{T}$ with positive measure for both signs suggests that the error term has genuine fluctuations of size $t^{1/4}$ infinitely often, not merely rare spikes."],"supporting_citations":[{"why":"Supplies the Voronoi-type expansion of $\\Delta(q_1q_2x;r_1,q_1,r_2,q_2)$ used as Lemma 3.1 and the high-moment bound (1.9) that Theorem 2.1 relies on.","marker":"[9]"},{"why":"Provides the kernel method and sign-change argument that Theorem 2.2 adapts to the congruence-condition error term.","marker":"[4]"},{"why":"Supplies the short-interval second-moment technique and the construction of many definite-sign subintervals used in Theorem 2.3.","marker":"[16]"},{"why":"Gives the estimates for the $R_{12},R_{21}$ and $G_{12},G_{21}$ sums used in Lemma 5.2 to bound the error term's short shifts.","marker":"[11]"},{"why":"Underwrites the mean value and mean-square estimates (1.7)-(1.8) used in Lemma 6.1 to force large measure for each sign.","marker":"[12]"},{"why":"States the Hilbert-type inequality used inside Lemma 5.2 to control off-diagonal contributions.","marker":"[14]"}],"fun_headline_variants":["Congruence divisor error flips sign in every √T window","Restricted divisor error: sign change in every √T interval","Congruence divisor error term: sign flips on √T scale","Moments of congruence divisor error match classical case","Divisor error under congruences: moments and √T-scale oscillation"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is the Voronoi-type expansion stated as Lemma 3.1 and taken without proof from the authors' earlier paper [9]: if that expansion, or the uniform error bounds inside it, fails, then the moment asymptotics, the sign-change theorems, and the $\\Omega$-result all collapse.","fun_headline_variants_meta":{"raw":{"variants":["Congruence divisor error flips sign in every √T window","Restricted divisor error: sign change in every √T interval","Congruence divisor error term: sign flips on √T scale","Moments of congruence divisor error match classical case","Divisor error under congruences: moments and √T-scale oscillation"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001822,"raw_usage":{"total_tokens":7349,"prompt_tokens":1305,"completion_tokens":6044,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":921,"completion_tokens_details":{"reasoning_tokens":5957}},"tokens_in":921,"tokens_out":6044,"duration_ms":45824,"temperature":1.0,"reasoning_tokens":5957,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T13:26:10.613164+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take one concrete pair, say $q_1=2,q_2=3$, and compute $\\Delta(q_1q_2t;r_1,q_1,r_2,q_2)$ at many points in a large interval $[T,T+C\\sqrt{T}]$. If for some large $T$ there is no point where the function, even after adding a perturbation $|f|\\le c_1 t^{1/4}$, takes both signs, Theorem 2.2 is false. A more targeted check is to evaluate Lemma 3.1 numerically for moderate $T$ and compare its right-hand side to the directly computed error term: an error larger than $O(\\log^3(q_1q_2T))$ outside the claimed bounds would invalidate the foundation of all four theorems.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the Voronoi-type expansion of $\\Delta(q_1q_2x;r_1,q_1,r_2,q_2)$ used as Lemma 3.1 and the high-moment bound (1.9) that Theorem 2.1 relies on."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the kernel method and sign-change argument that Theorem 2.2 adapts to the congruence-condition error term."},{"cited_title":"Sign changes of the err or term in Weyl’s law for Heisen- berg manifolds","cited_arxiv_id":null,"evidence_quote":"Supplies the short-interval second-moment technique and the construction of many definite-sign subintervals used in Theorem 2.3."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the estimates for the $R_{12},R_{21}$ and $G_{12},G_{21}$ sums used in Lemma 5.2 to bound the error term's short shifts."},{"cited_title":"M¨ uller and W","cited_arxiv_id":null,"evidence_quote":"Underwrites the mean value and mean-square estimates (1.7)-(1.8) used in Lemma 6.1 to force large measure for each sign."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"States the Hilbert-type inequality used inside Lemma 5.2 to control off-diagonal contributions."}],"review_version":1}