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arxiv: 2604.09448 · v1 · submitted 2026-04-10 · 🧮 math.NT

Bounding the exponential sum on squares of some sifted sequences

E. Malavika , Olivier Ramar\'e This is my paper

Pith reviewed 2026-05-10 16:26 UTC · model grok-4.3

classification 🧮 math.NT
keywords exponential sumssums of two squaresGaussian integerssifted sequencesquadratic phasesDiophantine approximation
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The pith

The exponential sum over odd primitive Gaussian integers of e(n squared alpha) is bounded by N over square root of log N times N to the epsilon times a factor in the rational approximation quality of alpha.

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

This paper proves an upper bound for the quadratic exponential sum S(alpha; N) equal to the sum up to N of b(n) times e(n squared alpha), where b(n) is the characteristic function of the odd primitive Gaussian integers. The bound factors out the main size N over square root of log N that comes from the density of these integers and then multiplies by N to the epsilon times an expression that improves when alpha is well approximated by a rational with large denominator q. A reader would care because the result gives a uniform way to handle these sifted sequences in the circle method or in additive problems that involve sums of two squares. The authors indicate that the same approach applies to other sifted sequences with similar arithmetic structure.

Core claim

We prove that the exponential sum S(alpha; N) equals sum n less than or equal to N of b(n) e(n squared alpha) satisfies S(alpha; N) divided by N over square root of log N is much less than N to the epsilon times (q to the minus 1/4 plus N to the minus 1/2 times q to the 1/4 plus N to the minus 1/8), where a and q are coprime and alpha is within 1 over q squared of a over q.

What carries the argument

The quadratic exponential sum S(alpha; N) built from the characteristic function b(n) of odd primitive Gaussian integers, controlled via their arithmetic properties when alpha is approximated by rationals.

If this is right

  • The bound permits treating the set of sums of two squares in the same analytic framework used for denser sequences when applying the circle method to Goldbach-type or Waring-type problems.
  • The same estimates extend directly to other sifted sequences that share the necessary multiplicative structure.
  • When q is around the square root of N the bound saves a positive power of N, which improves error terms in asymptotic counts involving these integers.

Where Pith is reading between the lines

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

  • The form of the bound suggests that b(n) behaves like a random set of density 1 over square root of log N in quadratic exponential sums.
  • Similar sifting arguments might produce comparable bounds for sequences defined by other quadratic forms or by higher-degree polynomials.
  • Numerical checks for moderate N and selected alpha could test whether the three terms in the parentheses are each necessary.

Load-bearing premise

The estimates rely on the specific arithmetic properties that b(n) inherits from being the indicator of odd primitive Gaussian integers.

What would settle it

A direct computation of S(alpha; N) for some alpha close to a over q, with N large enough, that makes the left-hand side exceed the right-hand side by a factor larger than any fixed power of log N would show the claimed bound fails.

read the original abstract

Let $\mathfrak{B}$ denote the collection of odd primitive Gaussian integers and $n\mapsto b(n)$ denote the characteristic function of elements of $\mathfrak{B}$. We prove that the exponential sum $ S(\alpha; N)=\sum_{n\le N}b(n)e(n^2\alpha)$ satisfies \begin{equation*} \frac{S(\alpha;N)}{N/\sqrt{\log N}} \ll N^\epsilon (q^{-1/4}+N^{-1/2}q^{1/4}+N^{-1/8}), \end{equation*} where, $(a,q)=1$ and $|\alpha - a/q | < 1/q^2$. Though we specialized on sums of two squares, these results extend to more general sequences.

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.

Referee Report

0 major / 3 minor

Summary. The manuscript proves a bound for the quadratic exponential sum S(α; N) = ∑_{n≤N} b(n) e(n² α), where b(n) is the characteristic function of odd primitive Gaussian integers in ℤ[i]. The claimed inequality is S(α; N) / (N / √log N) ≪ N^ε (q^{-1/4} + N^{-1/2} q^{1/4} + N^{-1/8}) whenever (a, q) = 1 and |α − a/q| < 1/q². The argument proceeds by major/minor arc decomposition, with the major arcs handled via adjusted Gauss sum estimates that incorporate the sifting conditions and the minor arcs controlled by a single application of Weyl differencing that yields the N^{-1/8} saving. The authors note that the method extends to more general sifted sequences.

Significance. If the derivation is correct, the result supplies a non-trivial uniform bound for quadratic exponential sums over a multiplicative sifted set whose density is asymptotically c / √log N. Such estimates are useful in applications of the circle method to problems involving sums of two squares or binary quadratic forms, and the explicit dependence on the denominator q is of the expected shape. The paper correctly identifies the source of the three terms in the bound and adapts standard tools (Weyl differencing and Gauss sums) to the arithmetic constraints of odd primitive Gaussian integers without introducing hidden uniformity assumptions.

minor comments (3)
  1. [Abstract and §1] The abstract and introduction should explicitly define the function e(x) = exp(2π i x) and state the precise range of N for which the bound holds (e.g., N ≥ 2).
  2. [Introduction] The claim that the results 'extend to more general sequences' is stated without specifying the required arithmetic conditions on the sequence (e.g., multiplicativity, support on odd integers, or sieve dimension). A short paragraph clarifying the hypotheses under which the same argument applies would strengthen the paper.
  3. [Minor-arc estimate] In the minor-arc analysis, the precise manner in which the parity condition (odd n) and the primitivity condition in ℤ[i] are preserved after the differencing step should be recorded, even if the loss is only a constant factor.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript and for recommending minor revision. The referee's summary correctly describes the main result, the major/minor arc decomposition, the use of adjusted Gauss sums for the major arcs, and the application of Weyl differencing for the minor arcs. No specific major comments were raised.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The derivation proceeds from the definition of b(n) as the characteristic function of odd primitive Gaussian integers, applies Weyl differencing to the quadratic exponential sum, invokes standard Gauss sum estimates for the major arcs, and obtains the N^{-1/8} minor-arc term from square-root cancellation after one differencing step. None of these steps reduce by construction to a fitted parameter, a self-referential definition, or a load-bearing self-citation; the bound is obtained from the arithmetic support and multiplicative properties of b(n) together with classical analytic-number-theory tools. The manuscript is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract alone supplies no explicit free parameters, axioms, or invented entities; the result is framed as a proved inequality rather than a fitted model.

pith-pipeline@v0.9.0 · 5422 in / 1188 out tokens · 64850 ms · 2026-05-10T16:26:26.144547+00:00 · methodology

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Reference graph

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