On the divisor problem with congruence conditions

Lirui Jia; Tianxin Cai; Wenguang Zhai

REVIEW 3 major objections 4 minor 18 references

On the divisor problem with congruence conditions

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read 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…

desk verdict 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. read the letter →

arxiv 1908.05598 v1 pith:4AUNCH2O submitted 2019-08-14 math.NT

Lirui Jia , Wenguang Zhai , Tianxin Cai This is my paper · ORCID

classification math.NT MSC 11N3711P21

keywords divisorproblemcongruenceconditionserrortermpowermomentssignchangesVoronoiformulashortintervals

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

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.

What carries the argument

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.

What would settle it

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.

Watch

Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

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.

Reading between the lines

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

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.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the 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.

What 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.

Soft 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.

Is 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.

Who 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.

Recommendation: 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.

Referee Report

3 major / 4 minor

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).

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 (3)

[§3, Lemma 3.1] 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.
[§3, proof of Theorem 2.1] 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.
[§4, proof of Theorem 2.2] 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.

minor comments (4)

[§5, Lemma 5.2] 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.
[§7] 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.
[§6, proof of Lemma 6.2] 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.
[§2, Theorem 2.1] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the derivation is a standard chain from an imported Voronoi-type lemma and prior moment bounds to new sign-change results, with no fitted parameter relabeled as a prediction.

full rationale

The paper's central claims are derived by substituting an external Voronoi-type formula (Lemma 3.1, quoted from the authors' earlier paper [9]) into smoothed local averages and estimating the resulting oscillatory sums. The target sign-change theorem (Theorem 2.2) is not equivalent to Lemma 3.1: Lemma 4.1 requires a nontrivial smoothing computation and error estimates, and the final sign-change conclusion is a separate step. The moment theorem (Theorem 2.1) assumes an upper bound of the form ∫|Δ|^{A0} ≪ T^{1+A0/4+ε} and derives asymptotics for lower moments; this is a conditional implication, not a circular definition. The constants C_k are explicit constants arising from the main term of the expansion, not fitted parameters, and the perturbation f(t) in Theorem 2.2 is arbitrary, so the sign-change assertion is not forced by construction. The fact that Lemma 3.1 and the bound (1.9) are cited from the authors' own prior work is not itself circular, because those cited results are independent tools with stated hypotheses and are not restatements of the new theorems. The reviewer's noted gaps—Lemma 3.1 is imported without proof, Theorem 2.1's proof is deferred to the approach of Liu [11], and the final sign-change argument after Lemma 4.1 is not written out—are concerns about verification and completeness, not about derivation-by-construction. No equation in the paper reduces to its own input, and no fitted quantity is renamed as a prediction.

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

The central claims depend on several results from the prior literature that are not proved in this paper. The most important is the Voronoi-type formula (Lemma 3.1) from the authors' own previous work [9], which is required for the moment asymptotic and all sign-change theorems. The paper also imports the moment bound (1.9) from [9] to set A0 = 262/27, Huxley's bound for Δ(x), and Hilbert's inequality as a standard tool. No free parameters are fitted; the constants in the theorems are absolute. No new entities are introduced.

assumptions (5)

domain assumption Huxley's estimate Δ(x) ≪ x^{131/416} log^{26947/8320} x
Used to derive the uniform bound (1.5) for Δ(x; r1,q1,r2,q2). Standard result cited as [5].
domain assumption Voronoi-type formula (Lemma 3.1) from [9]
Unproved in this paper; the central representation of Δ(q1q2x) as oscillatory sums plus error terms, load-bearing for all proofs.
domain assumption Moment bound (1.9) from [9]: ∫|Δ(q1q2x)|^A dx ≪ T^{1+A/4} L^{4A} for 0≤A≤262/27
Used to take A0 = 262/27 in Theorem 2.1 and Corollary 2.1.
domain assumption Müller and Nowak's mean value formulas (1.7) and (1.8)
Used in Lemma 6.1 to bound the first moment and the mean square.
standard math Hilbert's inequality (Lemma 5.1)
Used in the proof of Lemma 5.2 to estimate cross terms in the short-interval mean square.

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Pith. "Pith review of On the divisor problem with congruence conditions." pith.science (2026). https://pith.science/paper/4AUNCH2O

@misc{pith2026190805598,
  author       = {Pith},
  title        = {Pith review of: On the divisor problem with congruence conditions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4AUNCH2O}},
  note         = {Machine review of arXiv:1908.05598}
}

abstract

Let $d(n; r_1, q_1, r_2, q_2)$ be the number of factorization $n=n_1n_2$ satisfying $n_i\equiv r_i\pmod{q_i}$ ($i=1,2$) and $\Delta(x; r_1, q_1, r_2, q_2)$ be the error term of the summatory function of $d(n; r_1, q_1, r_2, q_2)$ with $x\geq (q_1q_2)^{1+\varepsilon}, 1\leq r_i\leq q_i$, and $(r_i, q_i)=1$ ($i=1, 2$). We study the power moments and sign changes of $\Delta(x; r_1, q_1, r_2, q_2)$, and prove that for a sufficiently large constant $C$, $\Delta(q_1q_2x; r_1, q_1, r_2, q_2)$ changes sign in the interval $[T,T+C\sqrt{T}]$ for any large $T$. Meanwhile, we show that for a small constant $c'$, there exist infinitely many subintervals of length $c'\sqrt{T}\log^{-7}T$ in $[T,2T]$ where $\pm \Delta(q_1q_2x; r_1, q_1, r_2, q_2)> c_5x^\frac{1}{4}$ always holds.

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Reference graph

Works this paper leans on

18 extracted references · 18 canonical work pages

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [9]

L. R. Jia and W. G. Zhai. A weighted divisor problem. Journal of Number Theory , 178:60 – 93, 2017

work page 2017

[2] [11]

K. Liu. On higher-power moments of the error term for the divisor problem with congruence conditions. Monatshefte f¨ ur Mathematik, 163(2):175–195, 2011

work page 2011

[3] [1]

Cram´ er

H. Cram´ er. ¨Uber zwei S¨ atze des Herrn G. H. Hardy. Mathematische Zeitschrift , 15(1):201– 210, 1922

work page 1922

[4] [2]

A. Good. Ein ω -Result at f¨ ur das quadratische Mittel der Riemannschen Ze tafunktion auf der kritischen Linie. Inventiones mathematicae , 41(3):233–251, 1977

work page 1977

[5] [3]

D. R. Heath-Brown. The distribution and moments of the er ror term in the Dirichlet divisor problem. Acta Arith, 60(4):389–415, 1992

work page 1992

[6] [4]

D. R. Heathbrown and K. Tsang. Sign changes of E (T), δ (x), and P (x). Journal of Number Theory, 49(1):73–83, 1994

work page 1994

[7] [5]

M. N. Huxley. Exponential sums and lattice points III. Proceedings of the London Mathe- matical Society, 87(03):591–609, 2003

work page 2003

[8] [6]

M. N. Huxley. Exponential sums and the Riemann zeta funct ion V. Proceedings of the London Mathematical Society, 90(01):1–41, 2005

work page 2005