arXiv:2603.00768v3

On bilinear sums with modular square roots and applications II

Stephan Baier

math.NT11L0511L0711L2611L4011J5411J7111K3611N35

Paper metadata

Original submission: February 28, 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Comments: This work has been merged into and superseded by arXiv:2605.01635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

In this article, we continue our recent investigations on bilinear sums and additive energies with modular square roots. Here we improve our recent results for the case when the ranges of variables are large. We use these results to make further partial progress on the large sieve for square moduli.

AI-generated audit

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

Theorem 2, Corollary 2, and Theorem 3 are correct under their stated hypotheses. A false prime-power character identification in the proof has a verified parity-aware repair that preserves the exponential-sum estimate and every downstream conclusion. The remaining findings are a parameter-rounding correction, a range correction in explanatory prose, and one notation typo; none changes a central result.

Theorem 2Correct

Bilinear sums with modular square roots for odd moduli

Pages 4 and 11–26 · Theorem 2 and Section 3 · arXiv:2603.00768v2

Weyl differencing and Poisson summation reduce the bilinear sum to the counting function $A(d)$ . After smoothing and evaluation of the quadratic Gauss sums, multiplicativity reduces the remaining complete sum to odd prime powers. Perel'muter's and Cochrane's estimates then give Lemma 2, including the factors from the squarefree and squarefull parts of $r$ . The parity defect in Equation (49) is repaired in the proof finding below: the corrected value of the character parameter has the same $p$ -adic order, so Equations (50) and (53), Lemma 2, and both bounds in Equation (5) are unchanged.

Full paper, version 2 ↗

Corollary 2Correct

Unweighted-phase specialization

Page 4 · Corollary 2 · arXiv:2603.00768v2

Setting $f\equiv0$ and $H=M$ in the first term of Theorem 2 gives exactly the displayed estimate when $1\leq M\leq r/2$ . The separate upper-range mismatch in the sentence after the corollary is an explanatory-scope defect and does not alter the corollary itself.

Full paper, version 2 ↗

Theorem 3Correct

Local density bound for Farey fractions with square denominators

Pages 6–7 and 26–29 · Theorem 3 and Section 4 · arXiv:2603.00768v2

The parametrization in Section 4.1 converts $P(b/r+z)$ into bilinear sums to which Theorem 2 applies. Balancing $L$ against the first and last terms gives Equation (78), while the cited elementary large-sieve estimate in Lemma 3 covers the complementary range. The two bounds overlap under $Q^{1/2+\varepsilon}\leq r\leq Q^{1-2\varepsilon}$ and yield the three terms stated in Theorem 3. Replacing the real choice of $H$ by its floor, as detailed below, changes only an absolute constant.

Full paper, version 2 ↗

02Proofs4 reported findingsContains incorrect or incomplete proofs

Equation (49) is false for every even prime-power exponent: the Jacobi symbol is then principal, not quadratic. A complete local repair preserves the prime-power estimate and all main results, but the printed proof is incorrect as written. The other findings are two harmless local formal corrections and one notation typo.

Equation (49)Incorrect as written · verified repair

The Jacobi character is principal when the exponent is even

Page 21 · Section 3.6.3, Equation (49) · arXiv:2603.00768v2

The proof sets $\chi=(\frac{\cdot}{p^m})$ and asserts that it is the unique character modulo $p^m$ of order $2$ , so that Cochrane's parameter is $c=\varphi(p^m)/2$ . This is correct only for odd $m$ . If $m$ is even, then $\chi(x)=(\frac{x}{p})^m=1$ on the units, so $\chi$ is principal and Definition (18) gives $c=\varphi(p^m)$ , not $\varphi(p^m)/2$ . The repair is to split according to the parity of $m$ . In both cases $\operatorname{ord}_p(c)=m-1$ . Thus, when $t\leq m-2$ , the $cg'(x)$ term in Equation (51), after division by $p^t$ , still vanishes modulo $p$ and the same linear-or-cubic critical-point equation and Equation (53) follow. When $t'=m-1$ , Proposition 4(ii) gives the displayed $O(p^{m-1/2})$ estimate for the residual sum independently of whether its multiplicative character is principal; $p=3$ is absorbed into the absolute constant. The case $t'\geq m$ remains trivial. Hence Equations (50) and (53), Lemma 2, and Theorems 2–3 all survive the correction.

Full paper, version 2 ↗

Equation (75)Minor formal correction

The differencing parameter must be integral

Pages 28–29 · Equations (74)–(75) and the subsequent parameter check · arXiv:2603.00768v2

Theorem 2 requires $H\in\mathbb N$ , whereas Equation (75) chooses the generally nonintegral value $H=1/(LF)$ . Replace it by $H=\lfloor1/(LF)\rfloor$ . The subsequent parameter check gives $1/(LF)\geq1$ , and for every $x\geq1$ one has $\lfloor x\rfloor\geq x/2$ . Therefore $1\leq H\leq\min\{1/(LF),M\}$ and every occurrence of $H^{-1/2}$ changes by at most the factor $\sqrt2$ . This is a verified local correction and leaves Equations (76)–(81) and Theorem 3 unchanged.

Full paper, version 2 ↗

Corollary 2 range discussionMinor formal correction

The advertised nontriviality range exceeds the corollary's hypotheses

Page 4 · Sentence immediately following Corollary 2 · arXiv:2603.00768v2

Corollary 2 assumes $1\leq M\leq r/2$ , but the following sentence advertises its bound as nontrivial and stronger than Equation (4) through the larger range $M\leq r$ . Replace that terminal upper bound by $M\leq r/2$ . No theorem, proof, or later application uses the extra interval $r/2<M\leq r$ .

Full paper, version 2 ↗

Section 3.6.1 setupTypo

The complete sum carries the wrong divisor parameter

Page 20 · First display in Section 3.6.1 · arXiv:2603.00768v2

The display begins with $\mathcal E_j(g_0,g_1,l_1,a\overline{q_i},\ldots;p_i^{m_i})$ , although Corollary 3 and the definition immediately preceding it use $g_4$ in that position, and the displayed formula for $F(x)$ also contains $g_4$ . Replace $g_1$ by $g_4$ . The intended parameter is unique and every subsequent calculation already uses it, so no estimate changes.

Full paper, version 2 ↗

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper: arXiv:2603.00768v3
Authors listed: Stephan Baier
Audit date: August 15, 2026

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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. 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P(delta)

is proved for every sufficiently small

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P

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P(delta)

holds for every

delta>0

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t>0

and subsequently applied at

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On bilinear sums with modular square roots and applications II

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

Bilinear sums with modular square roots for odd moduli

Unweighted-phase specialization

Local density bound for Farey fractions with square denominators

The Jacobi character is principal when the exponent is even

The differencing parameter must be integral

The advertised nontriviality range exceeds the corollary's hypotheses

The complete sum carries the wrong divisor parameter

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