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Zero distribution of power series and binary correlation of coefficients

T0 review · 0 major / 3 minor · reviewed 2026-05-24 · grok-4.3

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Pith's one-line read Zeros of power series with binary-correlated multipliers equidistribute according to a radial measure fixed by the coefficient magnitudes.

desk verdict The paper gives a conditional equidistribution result for zeros of power series with binary-correlated coefficients and verifies the setup on six explicit sequences. read the letter →

arxiv 2104.04812 v1 submitted 2021-04-10 math.CV math.PR

Jacques Benatar , Alexander Borichev , Mikhail Sodin This is my paper
classification math.CVmath.PR
keywords powerserieszerosbinarycorrelationsequidistributionradialmeasurescomplexanalysismultiplicativesequencesThue-MorsesequenceGolay-Rudin-Shapiro
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper studies zeros of power series sum ξ(n) a(n) z^n where a(n) is a smooth positive sequence and ξ(n) are complex multipliers. It shows that when a satisfies suitable smoothness conditions and ξ has binary correlations with a spectral gap, the zeros are equidistributed with respect to a radial measure determined by a alone. The result is applied to six families of multipliers: IID sequences, quadratic phases e(α n²) for Diophantine α, random multiplicative sequences, the Golay-Rudin-Shapiro sequence, the indicator of square-free integers, and the Thue-Morse sequence.

What carries the argument

Binary correlations of the multipliers ξ (two-point correlation function depends only on lag) together with a spectral gap condition; this reduces zero counting to an expectation over the correlation kernel.

What would settle it

Numerically compute the empirical zero distribution of the partial sums for the Thue-Morse sequence paired with a concrete smooth a, such as a(n) = n^{-1/2}, and compare against the predicted radial measure; systematic deviation would falsify the equidistribution claim.

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Extended reading notes

Core claim

Under assumptions on the smoothness of the sequence a and on the binary correlations of the multipliers ξ with no gaps in the spectrum, the zeroes of the power series are equidistributed with respect to a radial measure defined by the sequence a.

Load-bearing premise

The multipliers must possess binary correlations whose two-point function depends only on the lag, plus a spectral gap condition.

Editorial extensions

If this is right

  • The same radial equidistribution holds for IID multipliers under the stated smoothness on a.
  • It holds for quadratic phases e(α n²) when α is Diophantine.
  • It holds for the Golay-Rudin-Shapiro sequence.
  • It holds for the indicator function of the square-free integers.
  • It holds for the Thue-Morse sequence.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Binary correlations alone, without stronger mixing assumptions, suffice to control zero distribution for these series.
  • The technique may apply to other sequences whose correlation kernel admits a spectral gap, such as certain automatic sequences beyond those listed.
  • The radial measure depends only on a, suggesting that magnitude growth dominates zero placement once lag-dependent correlations are fixed.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

0 major / 3 minor

Summary. The manuscript claims that for power series f(z) = ∑ ξ(n) a(n) z^n with infinite radius of convergence, where a(n) is a smooth sequence of positive reals and ξ(n) is a sequence of complex multipliers possessing binary correlations (two-point correlation depending only on lag) together with a spectral gap, the zeros are equidistributed with respect to a radial measure determined solely by a, provided suitable smoothness assumptions hold on a. The reduction of zero-counting to an expectation over the correlation kernel is the key step. The result is applied to six concrete families: IID sequences, quadratic phases e(α n²) with Diophantine α, random multiplicative sequences, the Golay–Rudin–Shapiro sequence, the square-free indicator, and the Thue–Morse sequence.

Significance. If the central reduction is valid, the paper supplies a general mechanism linking binary correlation hypotheses on coefficients to equidistribution of zeros via a radial measure fixed by a alone. This unifies several previously separate examples under one set of hypotheses and supplies explicit, checkable instances (IID, quadratic phases, arithmetic sequences) where the correlation assumption can be verified directly. The approach avoids post-hoc parameter fitting and keeps the target measure independent of the correlation data.

minor comments (3)
  1. [Abstract] The abstract states that the radial measure is 'defined by the sequence a' but does not indicate in which section the explicit formula for the measure (presumably via the logarithmic potential or the associated density) is derived from a; a forward reference would help the reader locate the definition before the examples.
  2. [Assumptions paragraph] In the paragraph on assumptions, the spectral-gap condition on ξ is invoked to justify the reduction to the correlation kernel; a brief sentence clarifying whether this gap is used only for ergodicity or also for quantitative error bounds would make the logical dependence clearer.
  3. [Main theorem statement] The smoothness hypotheses on a are used to replace discrete sums by integrals; the precise modulus of continuity or decay rate required for the error term to be o(1) should be stated explicitly rather than left as 'certain assumptions on the smoothness'.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful summary of our results and for the positive evaluation of the significance of the central reduction from binary correlations to radial zero equidistribution. The recommendation of minor revision is noted; however, the report contains no specific major comments requiring clarification or modification.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity

full rationale

The central result is a conditional theorem: under external assumptions on smoothness of sequence a and on binary correlations plus spectral gap for multipliers ξ, the zeros of the power series ∑ ξ(n) a(n) z^n are equidistributed w.r.t. a radial measure that is defined directly from a. The derivation reduces zero-counting to an expectation over the correlation kernel once the hypothesis on ξ is granted; the smoothness conditions on a then justify integral approximations. No equation equates the target measure to a fitted quantity extracted from the same zeros, no self-citation chain is invoked to justify the uniqueness of the measure or the correlation hypothesis, and the listed examples are presented as cases where the ξ hypothesis can be checked independently. The derivation is therefore self-contained against the stated premises and does not reduce to its inputs by construction.

Assumptions & free parameters 0 free parameters · 2 assumptions · 0 invented entities

The central claim rests on the existence of binary correlations for ξ and smoothness of a; no free parameters or invented entities are mentioned in the abstract.

assumptions (2)
  • domain assumption ξ possesses binary correlations and has no gaps in the spectrum
    Invoked to reduce zero distribution to the radial measure of a.
  • domain assumption a is sufficiently smooth
    Required for the radial measure to be well-defined and for the equidistribution to hold.

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Pith. "Pith review of Zero distribution of power series and binary correlation of coefficients." pith.science (2026). https://pith.science/paper/2104.04812

@misc{pith2026210404812,
  author       = {Pith},
  title        = {Pith review of: Zero distribution of power series and binary correlation of coefficients},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2104.04812}},
  note         = {Machine review of arXiv:2104.04812}
}
abstract

We study the distribution of zeroes of power series with infinite radius of convergence. The coefficients of the series have the form $\xi(n)a(n)$, where $a$ is a smooth sequence of positive numbers, and $\xi$ is a sequence of complex-valued multipliers having binary correlations and no gaps in the spectrum. We show that under certain assumptions on the smoothness of the sequence $a$ and on the binary correlations of the multipliers $\xi$, the zeroes of the power series are equidistributed with respect to a radial measure defined by the sequence $a$. We apply our approach to several examples of the sequence $\xi$: (i) IID sequences, (ii) sequences $e(\alpha n^2)$ with Diophantine $\alpha$, (iii) random multiplicative sequences, (iv) the Golay--Rudin--Shapiro sequence, (v) the indicator function of the square-free integers, (vi) the Thue--Morse sequence.

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