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A sharp almost sure upper bound for partial sums of random multiplicative functions

T0 review · 2 major / 4 minor · reviewed 2026-08-03 · deepseek-v4-flash

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Pith's one-line read The paper proves that, almost surely, partial sums of Steinhaus and Rademacher random multiplicative functions are bounded by √x(log log x)^{1/4+ε} for every ε>0, matching the lower bound and settling the conjecture.

desk verdict Settles Harper's conjecture conditionally, but the load-bearing smoothing inequality is imported from an unpublished preprint without proof. read the letter →

arxiv 2607.29429 v1 pith:DVPYCFZX submitted 2026-07-31 math.NT math.PR

Benjamin Durkan , Andrew Pearce-Crump This is my paper · ORCID
classification math.NTmath.PR MSC 11N6411K6560G4260G44
keywords randommultiplicativefunctionsalmostsureboundsmartingaleshypercontractivitypartialsumsSteinhausmodelRademacherlawoftheiteratedlogarithm
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

This paper proves that, for either a Steinhaus or a Rademacher random multiplicative function, the partial sums up to x are almost surely bounded by √x times (log log x)^{1/4+ε} for every ε>0. Because a matching lower bound was previously known, this determines the sharp logarithmic exponent in both models and settles the conjecture on the size of large fluctuations. The proof decomposes the sum by the largest prime factor, reduces the problem to controlling the quadratic variation of a prime-indexed martingale, and obtains the required saving through a new conditional r-th moment estimate on a single prime block.

What carries the argument

The largest-prime decomposition writes M_f(x) as a y0-smooth sum, a prime-indexed martingale M^(1)_f(x) = ∑_{y0<p≤y_J} f(p)Ψ'_f(x/p,p), and a repeated-prime term (absent in the Rademacher model). The proof then uses a deterministic smoothing inequality, quoted from an unpublished preprint, to bound the quadratic variation of this martingale by a principal logarithmic mean square M_ℓ(x_i,y_j;f) plus auxiliary errors. The new device is the conditional block moment estimate E[U_j^r | F_{y_{j-1}}] ≪_r I_{j-1}^r, where U_j is the maximum over smoothness cutoffs in the j-th prime block and I_j is a tilted normalized Euler-product integral; together with a supermartingale tilt and a stopped concent

What would settle it

Verify the smoothing inequality (3.20) on the deterministic function f≡1 (or any fixed assignment of prime values) for large x and the final prime block. The theorem's proof would collapse if for some ε and unbounded x the quadratic variation V_ℓ(x;1)/x exceeded the claimed right-hand side by a factor growing with x.

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Extended reading notes

Core claim

The central assertion is that the almost sure upper bound for partial sums of random multiplicative functions is sharp: |M_f(x)| ≪_{ε,f} √x (log log x)^{1/4+ε} almost surely, for both the Steinhaus model (independent unit-circle phases) and the Rademacher model (independent signs on squarefree integers). Combined with a previously established lower bound, this fixes the exponent 1/4. The proof achieves this by showing that the principal logarithmic mean square, which dominates the quadratic variation of the largest-prime martingale, is controlled by a supermartingale of tilted Euler-product integrals; a fixed conditional r-th moment bound on one prime block replaces a first-moment estimate s

Load-bearing premise

The proof assumes a deterministic smoothing inequality, quoted from an unpublished preprint, that bounds the quadratic variation of the largest-prime martingale by the principal logarithmic mean square; if that inequality fails for the final prime block or under the strict smoothness convention, the quadratic variation estimate and hence the theorem would not follow.

Editorial extensions

If this is right

  • The logarithmic exponent 1/4 in the almost sure upper bound is sharp in both the Steinhaus and Rademacher models.
  • A law-of-the-iterated-logarithm-type statement follows: limsup_{x→∞} log(1+|M_f(x)|/√x)/log log log x = 1/4.
  • The fixed-order form (Proposition 1.1) provides almost sure bounds with any exponent 1/4+1/(2r)+η, potentially useful for later quantitative applications.
  • The method reduces the problem to a single-block moment estimate, which may be transferable to other random multiplicative structures or to character sums.

Reading between the lines

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

  • The conditional block moment technique could yield analogous sharp bounds for weighted partial sums or for higher moments of the martingale, since the loss in earlier approaches came entirely from summing first moments over many blocks.
  • If the quoted smoothing inequality were proven from scratch, the theorem would no longer depend on an unpublished preprint; if it fails on the final prime block, the main theorem would not follow.
  • A similar largest-prime decomposition with a conditional r-th moment step might apply to almost sure bounds for other random multiplicative objects, such as character sums or random Euler products at the critical line.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 4 minor

Summary. The paper proves that for both a Steinhaus and a Rademacher random multiplicative function f, and for every ε>0, almost surely |M_f(x)| ≪_{ε,f} √x (log log x)^{1/4+ε}. Together with Harper's almost sure lower bound, this indeed determines the logarithmic exponent and would settle Harper's conjecture. The proof follows Caich's largest-prime-factor decomposition and smoothing architecture, but replaces Caich's conditional first-moment step by a conditional r-th moment estimate on a single prime block (Proposition 5.2), which removes a √(log log x) loss. The main new technical content is the supermartingale tilt (Lemma 5.1), the fixed conditional block moment (Proposition 5.2), the combined stopping argument (Proposition 5.3), and the transfer to quadratic variation (Proposition 5.4). The Rademacher case is treated in Section 7 by parallel arguments with a shifted low-moment estimate.

Significance. If the proof is complete, the result is a major contribution: it settles the almost sure upper bound at the conjectured exponent 1/4 for both random multiplicative function models, matching Harper's lower bound. The paper's main new idea, a fixed conditional r-th moment estimate for one prime block, is natural and appears to give exactly the saving needed to compensate the union over the J ≍ log log x prime blocks. The exposition is generally careful, with explicit constants and many estimates verified in detail. The largest caveat is that the deterministic smoothing inequality at the heart of the argument is imported without proof from an unpublished preprint; see the major comments.

major comments (2)
  1. [§3.6, Proposition 3.15, Eq. (3.20)] The pointwise smoothing inequality (3.20) is the only bridge from the principal logarithmic mean squares M_ℓ to the quadratic variation V_ℓ of the largest-prime martingale. In the proof of Proposition 3.15 the authors state only that this inequality is 'the deterministic estimate obtained from [5, Section 7.1]' and give no derivation. This is load-bearing: Proposition 5.4 uses (3.20) to obtain V_ℓ(x_i;f)/x_i ≪ T(ℓ)L^{-1/2+1/r}, and the final martingale step (Lemma 3.4) then produces the claimed exponent. The same unproved inequality is transferred verbatim to the Rademacher case in Proposition 7.3. Because [5] is a preprint and the strict/weak smoothness conventions and final-prime-block range are precisely the points where a mismatch could occur, the paper should either prove (3.20) in an appendix or state it as a theorem with a complete proof, explicitly verifying the conventions used
  2. [§7.1, Proposition 7.3] The Rademacher analogue of the smoothing inequality is asserted by saying that 'the deterministic decomposition leading to (3.20) uses the largest-prime representation ... and deterministic inequalities for the resulting smooth sums,' and applying it to (7.2). This inherits the same gap as the Steinhaus case: the pointwise inequality (7.3) is not proved. In particular, the auxiliary estimates (3.21) are verified, but the principal inequality itself is not. Since the Rademacher theorem depends on (7.3) exactly as the Steinhaus theorem depends on (3.20), this is a second load-bearing unproved ingredient, not a model-specific technicality.
minor comments (4)
  1. [§3.3] The notation 'X_ℓ = exp(2ℓK)' is easy to misread. From the subsequent display 'log y_0 = 2^{ℓ^K - Kℓ^{K-1}}' it is clear that X_ℓ = exp(2^{ℓ^K}) and L=ℓ^K, but the superscript is not visible in the typeset formula. Please re-set these displays so that the exponent is unambiguous.
  2. [§3.6, Lemma 3.13] The range in Lemma 3.13 is stated uniformly for y_0 ≤ v ≤ y_J(1+1/X). It may be worth adding a short comment that the estimate remains valid when the interval (v/(1+1/X), v] extends beyond y_J, since the final block is truncated at x_i in the application.
  3. [§4, Lemma 4.1] In the quotation of Harper's estimate, the variable q is reused both as the exponent and later as a prime; although the local convention is clear, the notation is overloaded. A different symbol for the exponent would improve readability.
  4. [§5, proof of Proposition 5.2] The conditional Doob inequality is applied to the martingale q ↦ Ψ_f(z,q) along Q_j. The text does not explicitly justify that the index set Q_j, which contains y_{j-1} and primes, is well-ordered and that the endpoint y_{j-1} is included; this is true, but a brief comment would help the reader.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the proof is an analytic chain with all load-bearing imports from independent prior work; no fitted parameter is relabelled as a prediction.

full rationale

I walked the derivation chain: the interpolation over sparse test points (Lemma 3.8), the largest-prime decomposition (Lemma 3.9), the smooth and repeated-prime estimates (Lemmas 3.10–3.12), the smoothing inequality (Prop. 3.15), the Steinhaus low moment (Lemma 4.1), the conditional rth block moment (Prop. 5.2), the stopping argument (Prop. 5.3), the quadratic-variation transfer (Prop. 5.4), and the final martingale/Hoeffding step (Lemma 3.4), with the Rademacher analogues in Section 7. At no point is a quantity defined in terms of the target, and no parameter is fitted to data and then called a prediction. The principal imported estimate, Caich's deterministic smoothing inequality (3.20), is quoted from [5, Section 7.1] without proof; that is an external dependency and a potential correctness risk, but not circularity, because Caich is not an author of this paper and the inequality is not derived from the theorem being proved. Harper's lower bound is used only to state sharpness in Theorem 1/Corollary 1.2, not in the proof of the upper bound. Self-citation is absent: the reference list contains no work by Durkan or Pearce-Crump. The result therefore does not reduce by construction to its inputs.

Assumptions & free parameters 3 free parameters · 10 assumptions · 0 invented entities

Central claim rests on standard theorems (Shiu, Mertens, Parseval, hypercontractivity, martingale inequalities) and on specific estimates from Harper [9] and Caich [5]. The paper's own contribution is the conditional r-th moment estimate, which does not introduce fitted parameters or new postulated entities.

free parameters (3)
  • K = integer ≥ 100 with 2Kη > 10
    Proof parameter chosen so that union bounds over test points are summable; absorbed into implied constants.
  • c_0 = ≤ 10^{-3}, chosen sufficiently small (depends on A=1)
    Sparseness parameter for test points; chosen in Lemma 3.8 so that increments between test points are negligible.
  • m_0 = integer > 1/c_0
    Moment parameter in the smoothing estimates (D = 8m_0^2 - 8m_0 + 4); chosen to ensure convergence of auxiliary probability sums.
assumptions (10)
  • standard math Shiu's Brun–Titchmarsh theorem for multiplicative functions (used in Lemma 3.5)
    Gives uniform short-interval divisor-sum bounds; cited from [16].
  • standard math Mertens theorem with exponential error (Lemma 3.6)
    Primitive estimation of ∑_{p≤u} 1/p with error exp(-c√log u); from Montgomery–Vaughan [15].
  • standard math Parseval identity for Dirichlet series (Lemma 3.2)
    Transforms logarithmic mean squares of smooth sums into full-line Euler product integrals; from Montgomery–Vaughan [15, (5.26)].
  • standard math Bonami hypercontractive inequality for Steinhaus and Rademacher moments (Lemmas 3.1, 7.1)
    Provides 2m-th moment bounds in terms of weighted ℓ2 norms with divisor weights; cited from Bonami [4] via Caich [5].
  • standard math Doob's maximal inequalities (Lemma 3.3)
    Controls maxima of martingales and supermartingales; standard.
  • standard math Azuma–Hoeffding variant for complex martingale differences (Lemma 3.4)
    Gives exponential tail for the prime-indexed martingale under a predictable quadratic variation bound; cited from Caich [5, Lemma 3.12].
  • domain assumption Harper's low-moment estimate for Steinhaus Euler products (Lemma 4.1)
    States E[(1/logY ∫ |F_Y(1/2+it)/(1/2+it)|^2 dt)^{2/3}] ≪ (log logY)^{-1/3}; key to the stopping argument; proved in Harper [9].
  • domain assumption Harper's shifted low-moment estimate for Rademacher Euler products (Lemma 7.4)
    Provides the Rademacher analogue with a small-frequency omission; cited from Harper [9, Section 4.4].
  • domain assumption Caich's pointwise smoothing inequality (Proposition 3.15, eq. 3.20)
    Deterministic inequality bounding quadratic variation V_ℓ/x_i by principal mean squares plus auxiliary terms; quoted from Caich's preprint [5, Section 7.1] without proof here.
  • domain assumption Caich's block hypercontractive calculation for repeated-prime contributions (Lemma 3.10)
    Gives conditional moment and first-moment bounds for the M^{(2)} term; referenced from Caich [5, equations (13),(15)].

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Pith. "Pith review of A sharp almost sure upper bound for partial sums of random multiplicative functions." pith.science (2026). https://pith.science/paper/DVPYCFZX

@misc{pith2026260729429,
  author       = {Pith},
  title        = {Pith review of: A sharp almost sure upper bound for partial sums of random multiplicative functions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DVPYCFZX}},
  note         = {Machine review of arXiv:2607.29429}
}
abstract

We prove that, for either a Steinhaus random multiplicative function or a Rademacher random multiplicative function $f$, and every $\eps>0$, almost surely $$ \left|\sum_{n\le x}f(n)\right|\ll_{\eps,f}\sqrt{x}(\log\log x)^{1/4+\eps}. $$ Together with Harper's almost sure lower bound, this determines the sharp logarithmic exponent in both models. This settles Harper's conjecture on the large fluctuations of random multiplicative functions.

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Works this paper leans on

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