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arxiv: 2606.29040 · v1 · pith:DUL2URSInew · submitted 2026-06-27 · 🧮 math.NT · math.PR

Distribution of random multiplicative functions in short intervals, with proper normalization

Adam J. Harper , Kannan Soundararajan , Max Wenqiang Xu This is my paper

Pith reviewed 2026-06-30 08:19 UTC · model grok-4.3

classification 🧮 math.NT math.PR
keywords Steinhaus random multiplicative functionshort intervalslimiting distributionGaussian distributionpartial sumsnumber theory
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The pith

Partial sums of a Steinhaus random multiplicative function over short intervals converge to Gaussian after an adjusted normalization.

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

The paper studies sums of a Steinhaus random multiplicative function over intervals of length y where y tends to infinity yet remains o(x). It proves that these sums, after a suitable scaling, converge in law to a standard normal random variable for every such y. The required scaling coincides with the usual square root of y except when the interval length is close enough to x that arithmetic dependencies alter the variance. In contrast, when the interval length is comparable to x, no scaling produces a non-degenerate Gaussian limit.

Core claim

With an appropriate normalization the sum from n equals x to x plus y of f(n) converges in distribution to a standard Gaussian random variable, where the normalization factor differs from the square root of y precisely when y is very close to x.

What carries the argument

The variance-adjusted normalization of the partial sum, chosen so that the scaled sum has limiting standard normal distribution.

If this is right

  • The limiting Gaussian law holds uniformly across all regimes of y growing slower than x.
  • The effective variance of the sum is not always equal to y because of multiplicative dependencies.
  • Sums over intervals of length comparable to x cannot be made to converge to a non-degenerate Gaussian by any scaling.

Where Pith is reading between the lines

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

  • The change in normalization when y is close to x likely arises from the increasing influence of small prime factors on the correlations inside the interval.
  • Analogous variance adjustments may appear for other completely multiplicative random models, such as the Liouville function in short intervals.
  • Numerical checks of the predicted variance for y equal to x divided by log log x could be performed at moderate x to verify the transition.

Load-bearing premise

The random function is completely multiplicative with independent uniform phases on the unit circle at each prime, and the interval length y grows to infinity while remaining o(x).

What would settle it

Generate many independent realizations of the sum over [x, x+y] for a sequence of x tending to infinity and y equal to x to the power 0.99, apply the paper's normalization, and test whether the empirical distribution converges to the standard normal.

read the original abstract

We determine the limiting distribution of partial sums of a Steinhaus random multiplicative function $\sum_{x\le n \le x+y} f(n)$ over short intervals $[x, x+y]$, where $y \rightarrow \infty$ but $y=o(x)$. We show that with appropriate normalization, the limiting distribution is Gaussian for all such $y$. A key new feature of our result is that the normalization factor is different from the standard deviation $\sqrt{y}$ when $y$ is very close to $x$. In contrast, when $y \asymp x$ there is no normalization for which the limiting distribution is a non-degenerate Gaussian.

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

1 major / 0 minor

Summary. The manuscript claims to determine the limiting distribution of the partial sums ∑_{x≤n≤x+y} f(n) for a Steinhaus random multiplicative function f, where y→∞ but y=o(x). It asserts that with appropriate normalization the limiting distribution is Gaussian for every such y, with the normalization factor coinciding with √y except in a sub-regime where y is very close to x; when y≍x no normalization yields a non-degenerate Gaussian limit.

Significance. If the result holds, it supplies a complete description of Gaussian behavior for short-interval sums across the entire admissible range of y, isolating a transition in the normalizing constant near the scale of x. This complements the known non-Gaussian global behavior of ∑_{n≤X} f(n) and furnishes a precise, falsifiable statement about the local regime.

major comments (1)
  1. [Abstract] Abstract: the result is stated clearly but supplies no proof details, error bounds, or verification steps; the full manuscript text was not available for assessment, so the central claim cannot be verified from the given material.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their report. The primary concern raised is that the full manuscript text was unavailable for assessment, preventing verification of the central claims, and that the abstract lacks proof details, error bounds, or verification steps. We address this below. The complete paper, including all proofs and technical details, is publicly available on arXiv:2606.29040.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the result is stated clearly but supplies no proof details, error bounds, or verification steps; the full manuscript text was not available for assessment, so the central claim cannot be verified from the given material.

    Authors: The abstract is intentionally concise, as is conventional, to state the main theorem without technical details such as full proofs or explicit error bounds. The complete manuscript, which contains the full proofs, error bounds, and all verification steps for the limiting Gaussian distributions (including the transition in the normalizing constant when y is close to x), is available on arXiv under identifier 2606.29040. We apologize that the full text was not accessible during the initial review and would gladly supply any specific sections or clarifications upon request. We do not believe it is appropriate or necessary to expand the abstract with proof details. revision: no

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper states a theorem determining the limiting Gaussian distribution of normalized short-interval sums of a Steinhaus random multiplicative function, with an explicit adjustment to the normalization factor when y is close to x. No equations or steps reduce a claimed prediction to a fitted input by construction, nor does any load-bearing premise rest on a self-citation chain whose cited result itself depends on the target claim. The result is presented as a mathematical determination under stated hypotheses (y→∞, y=o(x)), consistent with known global behavior when y≍x, and is therefore self-contained against external probabilistic and analytic number theory benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Ledger is inferred from the abstract alone; the result rests on the definition of the Steinhaus model and the stated range of y.

axioms (2)
  • domain assumption Steinhaus random multiplicative function is completely multiplicative with f(p) uniform on the unit circle
    Stated in the abstract as the object of study.
  • domain assumption y → ∞ but y = o(x)
    The regime in which the limiting distribution is claimed to be Gaussian.

pith-pipeline@v0.9.1-grok · 5639 in / 1319 out tokens · 43107 ms · 2026-06-30T08:19:00.036879+00:00 · methodology

discussion (0)

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Reference graph

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