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REVIEW 3 major objections 3 minor 34 references

Unimodular Fake Mobius Functions

T0 review · 3 major / 3 minor · reviewed 2026-08-03 · deepseek-v4-flash

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Pith's one-line read Fake Möbius sums obey an explicit formula from zeta zeros, and their bias type is decided by Re(z+w).

desk verdict The explicit formula (15) is a genuinely new extension of Selberg–Delange, but Theorem 5.15 is false as stated: at z=0 every a_rho vanishes, so the bias classification collapses and the paper's own example gives persistent bias where it predicts apparent bias. read the letter →

arxiv 2512.18936 v3 pith:KVYVNLUS submitted 2025-12-22 math.NT

Ali Saraeb This is my paper · ORCID
classification math.NT MSC 11A2511M0611M2611N37
keywords fakeMöbiusfunctionsmultiplicativeSelberg–DelangemethodexplicitformulaRiemannzetafunctionnontrivialzerosbiaslogarithmicCesàromean
topics The Riemann Hypothesis
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 multiplicative functions whose prime-power values are arbitrary unit-modulus numbers assigned independently of the prime, and shows their Dirichlet series factor as ζ(s)^z ζ(2s)^w times a holomorphic Euler product. Assuming RH, simple zeta zeros, and a weak Mertens-type gap condition, it derives an explicit formula for the smoothed summatory function: a main term from s=1, a secondary term from s=1/2, an oscillatory sum over nontrivial zeros, and a small error. From this formula it derives an explicit bias criterion at scale x^{1/2}(Log x)^{w-1}: persistent bias if Re(z+w)>0, apparent bias if Re(z+w)=0, and unboundedness if Re(z+w)<0, with the constant c_{1/2}(z,w) determining the value. This extends the Selberg–Delange method to capture contributions from the critical line, and provides a tunable family of examples for prime-race-type phenomena.

What carries the argument

The central object is the zeta factorization F_f(s)=ζ(s)^z ζ(2s)^w G_f(s) with z=ε_1, w=ε_2-ε_1(ε_1+1)/2, together with the explicit formula that extracts, in addition to the s=1 main term, the secondary term from s=1/2 and a Laplace-integral term from each nontrivial zero ρ. Watson's lemma converts each such term into a power-of-log expansion, and the zero sum defines an almost-periodic function whose Fourier coefficients encode the bias.

What would settle it

Compute J_ρ(0)= (ρ-1)^{-z} G_f(ρ) Z_ρ(ρ) ζ(2ρ)^w Γ(ρ) for a chosen f and a few zeta zeros; if G_f(ρ)=0 for even one ρ, the pointwise-limit and unboundedness conclusions in Theorem 5.15 need re-examination. Separately, numerically evaluate B_f^exp(x) for Re(z+w)=0 with c_{1/2}≠0: if it converges to c_{1/2} as x→∞, the apparent-bias classification fails.

Watch

Extended reading notes

Core claim

For a multiplicative f with f(p^k)=ε_k, writing z=ε_1 and w=ε_2-ε_1(ε_1+1)/2, the paper proves (under RH, SZC, and a convergence condition) that A_f^exp(x) - Δ_1(x) = Δ_{1/2}(x) + Σ_ρ Δ_ρ(x) + O(x^a), where Δ_1, Δ_{1/2}, Δ_ρ are explicit Laplace integrals with Watson-type asymptotics. Consequently the normalized difference has limit c_{1/2}(z,w) when Re(z+w)>0, bounded logarithmic-Cesàro oscillation to c_{1/2} when Re(z+w)=0, and unbounded growth when Re(z+w)<0.

Load-bearing premise

The classification collapses if every Fourier coefficient a_ρ in the zero sum vanishes (equivalently, if G_f(ρ)=0 at every zeta zero for some admissible sequence), which the paper asserts is impossible without proof; the zero sum also requires an unproved lower bound on gaps between zeta zeros to converge.

Editorial extensions

If this is right

  • For completely multiplicative functions f(n)=ξ^{Ω(n)}, the bias type is explicitly determined by ξ alone, since (z,w) are polynomial in ξ; this yields an infinite family with persistent, apparent, or unbounded behavior at the square-root scale.
  • The expansion extracts contributions from the critical line Re(s)=1/2 that the classical Selberg–Delange method does not; the same machinery applies to any Dirichlet series of the form ζ(s)^z ζ(2s)^w G(s).
  • The bias criterion reduces to the known rules for ε_k ∈ {0,±1}, so earlier results for classical fake Möbius functions appear as special cases.
  • Higher coefficients ε_k (k≥3) enter only at a strictly lower order, so the two leading prime-power values determine the bias type.

Reading between the lines

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

  • A natural testable extension is to compute c_{1/2}(z,w) for periodic ε_k sequences and check whether it vanishes on a positive-measure surface in (z,w); the paper leaves the structure of its zeros unexplored.
  • The argument that the almost-periodic zero sum cannot converge to a limit unless all its Fourier coefficients vanish suggests the no-bias regime requires c_{1/2}(z,w)=0, but also presupposes at least one nonzero coefficient; a concrete check is evaluating G_f(ρ) for a zeta zero under a simple ε_k sequence.
  • The method likely transfers to Dirichlet L-functions or Dedekind zeta functions, where the analogue of the factorization would involve powers of L(s,χ), giving a new family of bias results for prime races.
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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

3 major / 3 minor

Summary. The paper introduces 'unimodular fake Möbius functions': multiplicative functions f:N→S^1∪{0} whose prime-power values are a prescribed sequence ε_k. Their Dirichlet series factor as F_f(s)=ζ(s)^z ζ(2s)^w G_f(s) with z=ε_1 and w=ε_2-ε_1(ε_1+1)/2, and G_f holomorphic/bounded on Re(s)>1/3. The main analytic claim is an explicit formula for A_f^exp(x)=∑ f(n)e^{-n/x}: subtracting the main contribution from s=1 leaves a secondary term from s=1/2, a sum over nontrivial zeta zeros, and an error. This is derived by contour integration with Hankel contours, followed by Watson-lemma expansions. The paper then defines persistent, apparent, and zero bias at the scale x^{1/2}(Log x)^{w-1} and states a criterion in Theorem 5.15, conditional on RH, SZC, and a weak Mertens-type zero-gap assumption (190).

Significance. The explicit formula, if correct, is a substantive extension of the Selberg-Delange method: it extracts critical-line contributions, is parameter-free in that z,w are read off from the first two coefficients, and reduces to the known integer cases in [19]. The detailed contour computation and the Watson-lemma analysis are valuable and largely coherent. However, the advertised bias classification is not valid as stated: the z=0 case is not excluded, and the proof relies on an unproved nonvanishing assertion for the zero Fourier coefficients. These are load-bearing for Theorem 5.15(ii)-(iii), so the paper needs a corrected and sharpened statement, not merely editorial revision.

major comments (3)
  1. [Theorem 5.15, Eq. (162), Eq. (311), proof after Lemma 6.17] The assertion 'a_ρ=0 iff J_ρ(0)=0, and the latter is impossible' is false when z=0. From Eq. (311), a_ρ = -sin(πz)/π Γ(1+z) λ_{ρ,0}, so sin(πz)=0 forces a_ρ=0 for every ρ even if J_ρ(0)≠0. Moreover Eq. (162) gives Δ_ρ(x)≡0 for z=0, so the zero-sum term S(x) in (198) is identically zero. Concretely, take ε_1=0, ε_2=i, ε_k=0 for k≥3; then z=0, w=i, Re(z+w)=0, and (198) yields B_f^exp(x;0,i)=c_{1/2}(0,i)+o(1) with c_{1/2}(0,i)≠0. The pointwise limit therefore exists: this is persistent bias, not apparent bias as Theorem 5.15(ii) claims. The same mechanism invalidates the unboundedness claim in (iii) for z=0. The theorem needs an explicit hypothesis z≠0 (plus at least one nonzero a_ρ), or a separate treatment of the z=0/integer cases as in Remark 5.16.
  2. [Eq. (163), Proposition 3.4, Lemma 6.17] Even away from z=0, the proof of (ii) and the unboundedness part of (iii) require that some Fourier coefficient a_ρ is nonzero, i.e. that J_ρ(0)≠0 for some zeta zero. The manuscript asserts this is 'impossible (see (163))', but (163) merely defines J_ρ; it gives no nonvanishing proof. J_ρ(0) contains the factor G_f(ρ), and Proposition 3.4 establishes only holomorphy and boundedness of G_f on Re(s)>1/3, not nonvanishing. Since G_f is an Euler product involving g_f(p^{-s}), a zero of g_f in the unit disk could make G_f(ρ)=0 for some ρ; this is not excluded by anything in the paper. The dichotomy in Lemma 6.17 is therefore not established. The bias classification should state an explicit nonvanishing hypothesis on the a_ρ, or prove one.
  3. [Abstract and Theorem 5.15, Assumption 5.11 / Eq. (190)] The abstract advertises the explicit criterion as obtained under RH and SZC only, but Theorem 5.15 also assumes the unproved weak-Mertens-type zero-gap condition (190). This assumption is load-bearing: it is used to prove convergence of the zero sum ∑_ρ Δ_ρ(x), the uniform boundedness condition (179), and hence the whole classification. The abstract and the statement of the advertised 'explicit criterion' should include this additional hypothesis, or the theorem should be reorganized so that the unconditional parts and the conditional parts are clearly separated.
minor comments (3)
  1. [Section 3.1] Typo: 'if 1≤n=∏_p p^{v_p(n)}, we define we define' should read 'we define'.
  2. [Figure 1.3.4 caption] Typo: 'shaded reigon' should be 'shaded region'.
  3. [Section 5 numerical illustrations] The figure captions are not fully self-contained; for exact reproduction it would help to list the coefficients ε_k and the truncation parameters used in (203).

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the explicit formula and bias classification are derived from a parameter-free zeta-factorization and contour integration, with no fitted inputs or self-citation loops.

full rationale

The derivation is self-contained. The parameters z = ε1 and w = ε2 − ε1(ε1 + 1)/2 are read off from the first two prime-power coefficients by equating power series (102)–(105); no parameter is fitted to the quantity being predicted. The factorization (14) is then used in a standard Perron/Mellin contour argument to produce the explicit formula (15), and the Laplace/Watson estimates in Theorems 5.1 and 5.7 are derived from local factorizations of ζ(s)^z and ζ(2s)^w (Propositions 2.8, 2.14, Corollary 2.16) rather than imported by citation. The bias definitions (Definition 5.13) are independent of the theorem’s conclusions, and Theorem 5.15’s classification follows analytically from (198) plus a genuine almost-periodicity argument for the zero sum. References to [19]/[20] are external benchmarks for integer cases, not load-bearing self-citations; the unproved zero-gap condition (190) is an openly stated assumption, not a disguised input. A separate mathematical gap at z = 0 — where sin(πz) makes all a_ρ vanish — is a correctness issue, not circularity.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central derivation uses no numbers fitted to data: z,w are determined by the first two coefficients ε1, ε2. The main extra assumptions are unproved analytic hypotheses (RH, SZC, zero-gap) plus an unstated nonvanishing condition on G_f at zeta zeros. No new physical or abstract entities are invented.

assumptions (5)
  • domain assumption Riemann Hypothesis (RH)
    Assumed throughout (Lemma 2.18, Theorem 5.1) to control log ζ on Re(s)>1/2.
  • domain assumption Simple Zeros Conjecture (SZC)
    Used to treat nontrivial zeros ρ as simple branch points; see §2.3 and the proof of Theorem 5.1.
  • domain assumption Weak Mertens hypothesis / zero spacing (190)
    Assumption 5.11, used to prove Σ Cρ <∞ and the uniform boundedness (179). The paper notes that only the zero-gap consequence (190), or even weaker bounds, is needed.
  • ad hoc to paper Nonvanishing of G_f at nontrivial zeros
    The proof of Theorem 5.15(ii)-(iii) requires at least one aρ≠0 and asserts Jρ(0)≠0 is impossible (see (163)), but G_f(ρ) may vanish for admissible ε_k; no argument is supplied.
  • standard math Analytic continuation of complex powers ζ(s)^z with chosen branch cuts
    Definitions 2.2-2.5 and 2.12-2.15; standard but nontrivial branch conventions needed for the contour computation.

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Cite this review

Pith. "Pith review of Unimodular Fake Mobius Functions." pith.science (2026). https://pith.science/paper/KVYVNLUS

@misc{pith2026251218936,
  author       = {Pith},
  title        = {Pith review of: Unimodular Fake Mobius Functions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KVYVNLUS}},
  note         = {Machine review of arXiv:2512.18936}
}
abstract

Let $\mathbb{S}^1$ denote the unit circle. We introduce and develop the analytic and bias theory of unimodular fake M\"obius functions, i.e. multiplicative functions $\mathfrak{f}:\mathbb{N} \to \mathbb{S}^1 \cup \{0\}$ whose prime-power values are prescribed by a fixed sequence $\{\varepsilon_k\}_{k\ge1}$ via the rule $\mathfrak{f}(p^k)=\varepsilon_k$ for every prime $p$ and every $k\ge1$. A key feature of these functions is that their Dirichlet series admit a factorization into complex powers of the Riemann zeta function. Our main analytic result is an explicit formula for the smoothed summatory function $\sum_{n\ge1}\mathfrak{f}(n)e^{-n/x}$, consisting of a leading main term together with a sequence of lower-order terms. The formula may be viewed as an extension of the Selberg-Delange method and is expected to be of independent interest. As an application, we introduce a notion of bias at a natural scale and obtain an explicit criterion distinguishing persistent bias, apparent bias, and no bias for unimodular fake M\"obius functions.

Figures

Figures reproduced from arXiv: 2512.18936 by the authors.

Figure 3.2
Figure 3.2. 1: Contour of integration ΓT 21 [PITH_FULL_IMAGE:figures/full_fig_p021_3_2.png] view at source ↗
Figure 5.1
Figure 5.1. (Finitely supported εk) : Take f(p) = ε1 = e iπ/5 , f(p 2 ) = ε2, and f(p k ) = 0 for k ≥ 3. Left/middle/right correspond to ε2 ∈ {1, − 1 4 +i √ 15/4, −1}, hence ℜ(z+w) = 1.25, 0, −0.75, illustrating the persistent, apparent, and unbiased regimes, respectively [PITH_FULL_IMAGE:figures/full_fig_p030_5_1.png] view at source ↗
Figure 5.2
Figure 5.2. (Completely multiplicative f) : Take εk = e ikθ for all k ≥ 0. Shown are three representative angles θ ∈ {π/5, π/3, 2π/3}, with ℜ(z + w) = 0.5590, 0, −0.5, illustrating persistent, apparent, and unbiased behaviour, respectively [PITH_FULL_IMAGE:figures/full_fig_p030_5_2.png] view at source ↗
Figures from the paper (1 more)
Figure 5.3
Figure 5.3. Figure 5.3: (Periodic εk) : Take the 2−periodic sequence ε2ℓ+1 = i and ε2ℓ+2 = −i, so that ℜ(z + w) = 1 2 > 0, in which case A exp f has a persistent bias c1/2(z, w) ≈ 0.0684338509001 + 0.1036422146372 i. 30 [PITH_FULL_IMAGE:figures/full_fig_p030_5_3.png]

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