arXiv:2603.25814v3

On bilinear sums with modular square roots and applications III

Stephan Baier

math.NT11L0511L0711L2611L4011J5411J7111K3611N35

Paper metadata

Original submission: March 26, 2026
Latest revision: July 20, 2026 · version 3
Claimed MSC: 11L05, 11L07, 11L26, 11L40, 11J54, 11J71, 11K36, 11N35
arXiv categories: math.NT
Paper license: arXiv non-exclusive distribution license ↗
Comments: This work has been merged into and superseded by arXiv:2605.01635

Abstract

We continue our investigations of bilinear sums with modular square roots and the large sieve for square moduli in our recent article "On bilinear sums with modular square roots and applications II", arXiv:2603.00768. In the present article, we focus on the case of prime square moduli for which our previous method in the said article did not yield any improvement. Now we modify this method to make progress for these moduli. The key idea is to restrict certain quadratic Gauss sums to reduced residue classes, which results in significant cancellations in certain cases.

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Audit summary

Audited against latest accessible manuscript v2; current record v3

Not a correctness certificate. A “Correct” result may include yellow typos or minor formal corrections that do not affect substantive soundness. It means this audit found no unresolved substantive error under the stated criteria; it does not replace expert scrutiny or formal verification.

Current report

Detailed mathematical audit

Generated August 15, 2026

01Statements3 reported findingsCorrect

The prime-square bilinear estimate, its unsmoothed corollary, and the resulting local large-sieve bound are supported for the stated parameter ranges. The reviewed mathematical source is arXiv version 2; the current arXiv record is version 3.

Theorem 4Correct

Bilinear sums with modular square roots modulo $p^2$

Pages 5 and 8–16 · Theorem 4 and Section 3 · arXiv:2603.25814v2

Weyl differencing and smoothing reduce the estimate to complete restricted quadratic Gauss sums. Proposition 3 removes the terms whose linear coefficient is divisible by $p$ , while Cochrane's critical-point estimate gives size $O(p)$ for simple critical points and $O(p^{3/2})$ only for the two exceptional residue classes of the dual variable. Averaging these exceptional classes yields Equation (27), and substitution into Equation (9) gives exactly the four terms in the theorem.

Corollary 1Correct

The unsmoothed bilinear bound

Pages 5–6 · Corollary 1 · arXiv:2603.25814v2

Taking $f=0$ and $H=M$ satisfies $H\leq\min\{1/(LF),M\}$ because $F=0$ . Substituting these values into Theorem 4 produces the displayed terms $L^{1/2}M^{1/2}r^{1/4}$ , $L^{1/2}r^{3/8}$ , and $M$ , with the same sequence norms and harmless $r^\varepsilon$ factor.

Theorem 3Correct

Local density bound for prime-square moduli

Pages 5 and 16–18 · Theorem 3 and Section 4 · arXiv:2603.25814v2

The standard conversion of $P(b/r+z)$ to the bilinear sum has $L=Q^{1+\varepsilon}r/\delta$ , $M\asymp Q^{1/2-\gamma}$ , and the stated derivative bound. Balancing the first and last terms gives $L=r^{1/2}Q^{-1/4-\gamma/2}$ . The checks of $L$ , $M$ , $H$ , and $\delta$ lead to Equation (42); the complementary elementary estimate covers both larger $\gamma$ and the remaining lower- $r$ range. Renaming $2\varepsilon$ as $\varepsilon$ gives the theorem's full interval and stated three-term bound.

02Proofs4 reported findingsCorrect

The Gauss-sum calculation, critical-point count, and parameter optimization are correct and complete. One displayed $p$ -adic order is a uniquely repairable typo and does not change any bound.

Proposition 3Correct and complete

Restricted quadratic Gauss sums modulo $p^2$

Pages 6–7 · Proposition 3 and Equation (7) · arXiv:2603.25814v2

Subtracting the contribution of multiples of $p$ from the ordinary quadratic Gauss sum gives every case in Equation (7). In particular, when $p\nmid a$ and $p\mid b$ , the two contributions are equal and cancel; when $p\mid a,b$ with $p^2\nmid a$ , Proposition 2 reduces the complete sum to a Gauss sum modulo $p$ and the removed contribution is exactly $p$ . These are the cases used in Section 3.

Section 3.4Typo

The displayed $p$ -adic order should be zero

Page 15 · Paragraph following Equation (26) · arXiv:2603.25814v2

The paper prints $t_p(f_{h,u,v})=\operatorname{ord}_p(f'_{h,u,v})=1$ . Replace $1$ by $0$ . After clearing the denominator, the derivative numerator has constant coefficient $-jhu^2$ , which is nonzero modulo $p$ under $(jhu,p)=1$ . The subsequent application of Proposition 4 already uses $t=0$ : a simple critical point contributes $p$ , while a triple critical point contributes $p^{3/2}$ . Thus the correction is unique and all downstream estimates are unchanged.

Section 3Correct and complete

Smoothing, Poisson summation, and critical-point averaging

Pages 9–16 · Equations (9)–(27) · arXiv:2603.25814v2

The decomposition according to divisibility by $p$ , both Poisson transformations, and the restricted Gauss-sum evaluations preserve the coprimality conditions. The derivative numerator is quartic. Its discriminant shows that multiple roots occur for at most two residue classes of $v$ modulo $p$ , and the second and third derivatives rule out multiplicity four for $p>3$ . Rapid Fourier decay then gives $O(p+p^{3/2}/H)$ for the inner average, which is sufficient for Equation (27).

Section 4Correct and complete

Optimization and complementary range

Pages 16–18 · Equations (28)–(43) · arXiv:2603.25814v2

The choice $H=(LF)^{-1}$ lies in the permitted range after Equation (41), and balancing determines a $\delta$ lying in its allowed interval. Equation (40) covers $\gamma\leq1/2$ in the indicated range; Lemma 2 gives $O((1+Qr^{-1})Q^{2\varepsilon})$ outside it. Since $Qr^{-1}\leq Q^{5/8}r^{-1/4}$ for $r\geq Q^{1/2}$ , the two estimates join without a gap.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper: arXiv:2603.25814v3
Authors listed: Stephan Baier
Audit date: August 15, 2026

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You are a mathematical auditor. Evaluate only the correctness of the paper's central statements, the correctness and completeness of its proofs, and evidence-backed non-novelty. Do not make editorial comments or evaluate importance, exposition, style, or suitability for publication. Keep the report neutral, precise, and professional. PART 1 — CORRECTNESS OF THE MAIN STATEMENTS Identify the paper's central statements and assess each one separately using exactly one of these outcomes: - Correct: the statement has a correct, verifiable proof; or, despite a defect in the printed proof, you can give a fully verified repair or independent proof. - Incorrect: you can give strong formal evidence that the statement as written is false, preferably a counterexample or a complete contradiction. - Not able to verify: the supplied proof does not establish the statement, and you can neither supply a fully verified proof nor disprove it. 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If the list is empty, the section may say only “No non-novelty findings.” PART 4 — TECHNICAL AND EVIDENTIARY RULES 1. Criticism must never be overstated. Call a statement incorrect, a proof incorrect or incomplete, or a result non-new only when strong formal evidence supports that exact judgment. Otherwise use Not able to verify where that outcome exists, or remain silent in the novelty section. 2. Review a specified paper version. Give exact page, theorem, equation, or proof locations; restate relevant quantifiers and hypotheses; and check domains, endpoints, parameter ranges, constants, and dependencies explicitly. Apply a defect only to the regimes and downstream results that actually depend on it. 3. Immediate-consequence closure: when deciding whether a conclusion is supported, treat the argument as including every immediate formal consequence of facts established earlier in the paper and available at that point. An immediate formal consequence is a local, mechanically checkable inference using only definitions already in force, elementary logic, direct substitution or renaming, weakening of an inequality or set inclusion, or a monotonicity or symmetry property explicit in a definition or already proved. 4. Such an inference must introduce no new lemma, estimate, construction, case analysis, limiting or compactness argument, external result, or new issue of uniformity or parameter dependence. Check the direction of monotonicity and the dependence of every threshold or constant explicitly. Do not exchange pointwise and uniform conclusions, change exceptional sets, interchange quantifiers or limits, or extend an endpoint under this rule unless the required fact was already established. 5. If the claimed conclusion follows after immediate-consequence closure, it is fully supported. Do not report a gap, unsupported statement, minor formal correction, or recommendation to add an explanatory sentence, and do not mention the suppressed item in the report. This suppression does not apply to a genuine typographical error as defined below, because a typo is a literal notation defect that still requires a textual correction. For example, if

P(delta)

is proved for every sufficiently small

delta>0

and

P

is immediately monotone in the required direction, then the claim that

P(delta)

holds for every

delta>0

requires no correction; the omitted monotonicity step is exposition rather than an audit finding. 6. Immediate-consequence closure permits reconstruction of an omitted inference; it does not permit silent alteration of the printed domain, hypotheses, or quantifiers of an intermediate statement later invoked outside its stated scope. If formal validity requires changing the text of that statement or its stated range, report the appropriate finding even when the repair is easy. Thus a result stated only for

t>0

and subsequently applied at

t=0

remains a formal scope mismatch unless the endpoint case was separately established before the application. 7. Apply this mandatory gate before recording any adverse, unverified, typographical, or minor formal finding: (a) identify the exact assertion or inference alleged to be defective; (b) reconstruct the strongest conclusion following from the preceding argument after all admissible immediate formal consequences are added; (c) identify the specific defect that remains — a formally demonstrated falsehood, a concrete nontrivial unresolved proof obligation, or a genuine domain, hypothesis, quantifier, or notation mismatch requiring a textual correction; and (d) state the evidence and minimal repair. If no specific defect remains, or if the only possible addition would be explanatory, suppress the finding entirely. Use Not able to verify only when a precise, nontrivial unresolved obligation can be identified. 8. A typographical error is an incorrect symbol, index, exponent, sign, variable name, or notation mismatch whose unique intended correction is mechanically determined from the surrounding text and does not alter the mathematical argument. Report it in yellow as Typo, with the printed text, correction, exact location, and reason it is harmless. It does not lower an overall status. 9. A minor formal correction is a boundary, endpoint, or displayed parameter-range defect that can be repaired consistently in every affected statement by a local change, without a new argument, without invalidating any related result or application, and without creating another substantive defect. Report it in yellow as Minor formal correction, with the same details. It does not lower an overall status. 10. Do not use either minor category if more than one materially different repair is possible; if the repair changes the substance or scope of a central result; if an excluded case is used later; if the correction fails to propagate consistently; or if a new mathematical idea is required. In such cases use the strongest substantive classification actually supported by evidence. 11. Write every mathematical expression as valid, compilable LaTeX. Use

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...

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On bilinear sums with modular square roots and applications III

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

Bilinear sums with modular square roots modulo p2p^2p2

The unsmoothed bilinear bound

Local density bound for prime-square moduli

Restricted quadratic Gauss sums modulo p2p^2p2

The displayed ppp-adic order should be zero

Smoothing, Poisson summation, and critical-point averaging

Optimization and complementary range

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Bilinear sums with modular square roots modulo $p^2$

Restricted quadratic Gauss sums modulo $p^2$

The displayed $p$ -adic order should be zero