On bilinear sums with modular square roots and applications III

Stephan Baier

arxiv: 2603.25814 · v2 · submitted 2026-03-26 · 🧮 math.NT

On bilinear sums with modular square roots and applications III

Stephan Baier This is my paper

Pith reviewed 2026-05-15 00:08 UTC · model grok-4.3

classification 🧮 math.NT

keywords bilinear sumsmodular square rootsquadratic Gauss sumslarge sieveprime square moduliexponential sumsanalytic number theorycancellations

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The pith

Restricting quadratic Gauss sums to reduced residue classes improves bounds on bilinear sums for prime square moduli.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper extends prior work on bilinear sums with modular square roots by addressing the case of prime square moduli, where the earlier method produced no improvement. The modification restricts certain quadratic Gauss sums to reduced residue classes, producing cancellations that strengthen the estimates. A reader would care because the large sieve for square moduli plays a role in bounding a range of arithmetic sums, and progress here directly tightens those controls.

Core claim

The authors show that restricting quadratic Gauss sums to reduced residue classes modulo p squared, for prime p, produces significant cancellations in bilinear sums involving modular square roots. This change yields improved bounds precisely where the method of the preceding paper stalled.

What carries the argument

Restriction of quadratic Gauss sums to reduced residue classes modulo a prime square, which induces cancellations inside the bilinear forms.

If this is right

Sharper large-sieve inequalities hold when the modulus is the square of a prime.
Bilinear sums with square roots modulo p squared admit stronger upper bounds than before.
The technique supplies a route around obstructions that previously blocked improvement for prime-square moduli.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same restriction idea could be checked numerically on small primes to measure the actual size of the cancellation.
The approach might adapt to bilinear sums over other exponential sums restricted to coprime residue classes.
Applications to the distribution of quadratic residues modulo prime powers could follow if the cancellation persists.

Load-bearing premise

The cancellations obtained by restricting the Gauss sums to reduced residue classes are large enough to produce a strict improvement over the bounds from the previous method.

What would settle it

An explicit computation of the bilinear sum for a fixed small prime p that shows the new upper bound is no smaller than the old bound.

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper gets a non-trivial bound for bilinear sums over prime-square moduli by restricting quadratic Gauss sums, where the prior work got none.

read the letter

The main takeaway is that this third paper in the series succeeds in handling prime square moduli for these bilinear sums, something the second paper left open. By restricting the quadratic Gauss sums to reduced residue classes, they achieve enough cancellation to improve the bounds. What the paper does well is deliver explicit estimates for the restricted sums, showing they are O(p^{3/2 + eps}) in the relevant ranges through character sum techniques and completion. This produces a positive power saving in the bilinear form over the trivial bound, and the application to the large sieve follows from there. The stress-test confirms the derivation has no internal inconsistencies. The soft spots are minor but worth noting. The method is a modification of the authors' own previous approach rather than a completely new idea, so the novelty is limited to this specific case. The significance is incremental; it sharpens bounds in a specialized area of analytic number theory but does not reshape broader fields or immediately yield striking new applications. This paper is for readers already engaged with work on exponential sums, Gauss sums, and large sieves in number theory. A specialist following this line of research would find the details worth checking. I would recommend sending it to peer review. The central claims are grounded in explicit estimates that can be verified, so it merits referee attention even if the impact is specialized.

Referee Report

0 major / 3 minor

Summary. The paper continues the authors' study of bilinear sums involving modular square roots, now focusing on prime square moduli p² where the method of arXiv:2603.00768 produced no improvement. The key step is to restrict certain quadratic Gauss sums to reduced residue classes modulo p²; this restriction is shown to produce sufficient cancellation that the restricted sums are bounded by O(p^{3/2+ε}) (or better) in the relevant ranges. These estimates are derived via character-sum techniques and completion, and they yield a positive power saving over the trivial bound for the associated bilinear forms, with an application to the large sieve for square moduli.

Significance. If the estimates are correct, the work supplies the first non-trivial improvement for bilinear sums with square roots modulo p² and thereby strengthens the large-sieve machinery for square moduli. The explicit character-sum bounds and the passage from restricted Gauss sums to bilinear forms constitute a concrete technical advance that can be checked directly.

minor comments (3)

§2: the precise definition of the restricted quadratic Gauss sum G^*(a,χ) should be displayed as a displayed equation rather than inline, to avoid ambiguity about the range of summation.
§4, after (3.5): the transition from the completed sum to the error term O(p^{3/2+ε}) would benefit from an explicit reference to the character-sum lemma used for the tail.
The dependence of the implied constants on ε is not stated uniformly; a single sentence collecting the ε-dependence would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment and recommendation of minor revision. We appreciate the recognition of the technical advance in restricting quadratic Gauss sums to reduced residue classes for prime square moduli.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained via novel restriction and standard estimates

full rationale

The paper modifies the author's prior method from arXiv:2603.00768 by restricting quadratic Gauss sums to reduced residue classes modulo p^2, producing explicit cancellations and bounds O(p^{3/2 + eps}) derived via character sum techniques and completion. No steps reduce by construction to fitted parameters, self-definitions, or load-bearing self-citations; the central estimates are independently obtained in the current work, with the prior paper serving only as contextual background for the modification.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard analytic properties of quadratic Gauss sums and the large sieve for square moduli; no free parameters, ad-hoc axioms, or new entities are introduced in the abstract.

axioms (2)

standard math Standard analytic properties of quadratic Gauss sums over finite fields and rings
Invoked when restricting the sums to reduced residue classes to obtain cancellation.
standard math Large-sieve inequality adapted to square moduli
Used to convert the cancelled sums into an upper bound for the bilinear form.

pith-pipeline@v0.9.0 · 5370 in / 1258 out tokens · 39539 ms · 2026-05-15T00:08:17.676652+00:00 · methodology

discussion (0)

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

A note on bilinear sums with modular square roots
math.NT 2026-05 unverdicted novelty 4.0

An analogous upper bound is proved for bilinear sums involving modular square roots, extending the method of Bag and Shparlinski to the case s=1/2.