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arxiv: 2605.19105 · v1 · pith:TLGIIM4Qnew · submitted 2026-05-18 · 🧮 math.NT

Hal\'asz theorems for Gaussian ideals in sectors and short intervals

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classification 🧮 math.NT
keywords Halász theoremGaussian integersmultiplicative functionspretentious distanceangular sectorsshort intervalsidealsnorm compression
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The pith

A quantitative Halász theorem holds for multiplicative functions on the nonzero ideals of the Gaussian integers, including sectorial and short-interval analogues.

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

This paper proves Halász theorems for multiplicative functions on the ideals of Z[i]. It establishes that the sum of such a function over ideals in a fixed sector is asymptotically the proportion of the total sum when the function is angularly non-pretentious. It further shows a short-interval version in annular sectors with growing radial thickness under an additional non-degeneracy condition. These extensions allow control of sums in the complex plane using distance to characters of the form N to the power it, extending classical results from the rational integers.

Core claim

We prove a quantitative Halász theorem for multiplicative functions on the nonzero ideals of Z[i], with bounds controlled by pretentious distance to the Archimedean characters N^{it}. Under angular non-pretentiousness, the sum of f over ideals lying in a fixed sector is asymptotically given by the expected proportion of the unrestricted sum. Under angular non-pretentiousness and a non-degeneracy condition on conjugate prime pairs, we prove a sectorial short-interval version of the Halász theorem for annular sectors whose radial thickness tends to infinity.

What carries the argument

Pretentious distance to the Archimedean characters N^{it}, together with angular Fourier expansion and norm-compression to functions on the natural numbers.

If this is right

  • The sum over ideals in a fixed sector equals the expected proportion of the total sum asymptotically.
  • A sectorial short-interval Halász theorem holds for annular sectors with radial thickness going to infinity.
  • Bounds on the sums are controlled quantitatively by the pretentious distance.
  • The proofs reduce the problem to the classical setting via norm compression and Mangerel's theorem.

Where Pith is reading between the lines

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

  • This framework may extend to other rings of integers in quadratic fields with suitable modifications to the angular conditions.
  • The non-degeneracy condition on conjugate prime pairs could be checked explicitly for standard functions such as the Möbius function.
  • These results open the door to studying the distribution of prime ideals in angular sectors with short-interval control.

Load-bearing premise

The sectorial short-interval result assumes both angular non-pretentiousness and a non-degeneracy condition on conjugate prime pairs, plus the validity of Mangerel's theorem after norm-compression.

What would settle it

Finding a multiplicative function on Z[i] ideals that is angularly non-pretentious and satisfies the non-degeneracy condition but for which the sector sum deviates significantly from the expected proportion would falsify the claim.

read the original abstract

We prove a quantitative Hal\'asz theorem for multiplicative functions on the nonzero ideals of $\mathbb{Z}[i]$, with bounds controlled by pretentious distance to the Archimedean characters $N^{it}$. We also prove a sectorial analogue: under angular non-pretentiousness, the sum of $f$ over ideals lying in a fixed sector is asymptotically given by the expected proportion of the unrestricted sum. Finally, under angular non-pretentiousness and a non-degeneracy condition on conjugate prime pairs, we prove a sectorial short-interval version of the Hal\'asz theorem for annular sectors whose radial thickness tends to infinity. The proof of the sectorial short-interval Hal\'asz theorem uses angular Fourier expansion, norm-compression to multiplicative functions on $\mathbb{N}$, and a theorem of Mangerel.

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

2 major / 2 minor

Summary. The manuscript proves a quantitative Halász theorem for multiplicative functions on the nonzero ideals of ℤ[i], with bounds controlled by pretentious distance to the Archimedean characters N^{it}. It establishes a sectorial analogue: under angular non-pretentiousness, the sum of f over ideals in a fixed sector is asymptotically the expected proportion of the unrestricted sum. Under angular non-pretentiousness plus a non-degeneracy condition on conjugate prime pairs, it proves a sectorial short-interval Halász theorem for annular sectors whose radial thickness tends to infinity. The proof of the short-interval result proceeds via angular Fourier expansion of the sector indicator, norm-compression of the multiplicative function on ideals to a function on ℕ, and an application of Mangerel's theorem.

Significance. If the central claims hold with the stated error terms, the work extends classical Halász theorems from ℤ to the Gaussian integers, incorporating angular sectors and short radial intervals. The norm-compression reduction combined with Mangerel's theorem is a technically interesting strategy that could be useful for distribution problems in quadratic fields. The paper receives credit for outlining a coherent proof architecture that reduces the ideal setting to the rational integers while preserving the pretentious-distance control.

major comments (2)
  1. [Abstract and proof of short-interval theorem] The sectorial short-interval theorem (stated in the abstract and presumably proved in the final section) invokes norm-compression after angular Fourier expansion. Norm-compression maps each ideal to its norm and therefore discards argument information; when the annular sector has radial thickness tending to infinity but still o of the radius, the manuscript must supply an explicit uniformity statement or error bound showing that the discrepancy between the compressed sum and the angularly localized sum is o of the main term. No such bound is indicated in the abstract or the reader's summary of the argument.
  2. [Statement of short-interval theorem] The non-degeneracy condition on conjugate prime pairs is listed as an additional hypothesis for the short-interval result. The manuscript should verify that this condition is both necessary and sufficient to control the angular error after compression; if the condition is only used to rule out a specific cancellation, an explicit counterexample or quantitative estimate showing the error remains controllable when the condition holds would strengthen the claim.
minor comments (2)
  1. [Introduction] The definition of angular non-pretentiousness should be stated explicitly with a formula (e.g., involving the pretentious distance in a fixed angular range) at the first appearance rather than left implicit from the classical case.
  2. [References and proof of short-interval result] Ensure that the reference to Mangerel's theorem includes the precise citation and states the exact form of the theorem applied after norm-compression.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive suggestions regarding the short-interval theorem. We address the two major comments below and indicate planned revisions to improve clarity and completeness.

read point-by-point responses

  1. Referee: [Abstract and proof of short-interval theorem] The sectorial short-interval theorem (stated in the abstract and presumably proved in the final section) invokes norm-compression after angular Fourier expansion. Norm-compression maps each ideal to its norm and therefore discards argument information; when the annular sector has radial thickness tending to infinity but still o of the radius, the manuscript must supply an explicit uniformity statement or error bound showing that the discrepancy between the compressed sum and the angularly localized sum is o of the main term. No such bound is indicated in the abstract or the reader's summary of the argument.

    Authors: We agree that an explicit uniformity statement would strengthen the presentation. In Section 5 the angular Fourier expansion of the sector indicator is truncated with an error controlled by the decay of coefficients, after which norm-compression is applied termwise; the resulting discrepancy is absorbed into the o(main term) by combining the quantitative bound from Mangerel's theorem with the angular non-pretentiousness hypothesis. To make this transparent we will revise the abstract to mention that the error after compression is o of the main term uniformly in the stated range, and we will add a short remark in the introduction summarizing the error estimate. revision: yes

  2. Referee: [Statement of short-interval theorem] The non-degeneracy condition on conjugate prime pairs is listed as an additional hypothesis for the short-interval result. The manuscript should verify that this condition is both necessary and sufficient to control the angular error after compression; if the condition is only used to rule out a specific cancellation, an explicit counterexample or quantitative estimate showing the error remains controllable when the condition holds would strengthen the claim.

    Authors: The non-degeneracy condition prevents resonant phase cancellations arising from conjugate prime pairs under norm-compression. We will add a brief discussion (new Remark 5.3) that sketches a counterexample multiplicative function violating the condition for which the angular error fails to be o(main term). When the condition holds, the proof already shows that any residual angular discrepancy is dominated by the pretentious-distance term supplied by Mangerel's theorem, yielding the claimed o(main term) bound. This addition will make the role of the hypothesis explicit. revision: yes

Circularity Check

0 steps flagged

No circularity: proofs rely on external Mangerel theorem and standard techniques

full rationale

The derivation chain proceeds from pretentious distance hypotheses and angular non-pretentiousness to asymptotic statements via angular Fourier expansion of sector indicators, norm-compression of ideal multiplicative functions to ℕ, and direct invocation of Mangerel's theorem. These steps are independent of the paper's own fitted quantities or self-referential definitions; the short-interval annular sector result is stated as a theorem under explicit non-degeneracy conditions on conjugate primes rather than reducing by construction to its inputs. No self-citation load-bearing steps, uniqueness theorems imported from the same authors, or ansatzes smuggled via prior work appear in the abstract or described proof outline. The central claims therefore remain self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The proofs rest on standard analytic number theory axioms (properties of multiplicative functions, definition of pretentious distance to N^{it}, existence of angular Fourier expansions) and the external theorem of Mangerel. No free parameters or invented entities are introduced in the abstract; the non-degeneracy condition on conjugate prime pairs is an additional domain assumption required for the short-interval result.

axioms (2)
  • domain assumption Multiplicative functions on ideals of ℤ[i] admit a pretentious distance to Archimedean characters N^{it} that controls mean values.
    Invoked to obtain the quantitative bounds in the main Halász theorem.
  • standard math Angular Fourier expansion and norm compression reduce the sectorial problem to ordinary multiplicative functions on ℕ.
    Used in the proof of the sectorial short-interval version.

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11 extracted references · 11 canonical work pages · 1 internal anchor

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