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On the value distribution of the Riemann zeta-function with general shifts

T0 review · 3 major / 5 minor · reviewed 2026-08-01 · deepseek-v4-flash

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Pith's one-line read This paper proves that the Riemann zeta-function is jointly universal under shifts by powers of log t, and gives explicit discrepancy rates for the Bohr-Jessen limit theorem under very general shifts.

desk verdict Genuinely new results for log-power shifts, with a concrete (repairable) approximation gap in Lemma 6.5 and an underproved Theorem 1.6; deserves peer review. read the letter →

arxiv 2607.20041 v1 pith:GGT5HH2A submitted 2026-07-22 math.NT

Keita Nakai This is my paper · ORCID
classification math.NT MSC 11M0611M2611M4160F05
keywords Riemannzeta-functionuniversalityBohr-Jessenlimittheoremdiscrepancylogarithmicshiftsvaluedistributionzero-densityestimatesDirichletpolynomials
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 is trying to establish how much of the zeta-function's value distribution survives when the usual vertical shift it is replaced by a slowly growing shift iγ(t). For a broad class of fast-growing shifts—polynomial, exponential, and beyond—it proves a quantitative Bohr-Jessen discrepancy bound of order (log γ(T))^{-σ}; for the previously unreachable logarithmic-power shifts γ(t)=(log t)^α with α>1, it proves joint universality and a quantitative Bohr-Jessen result. If correct, the conclusion is that logarithmic-power shifts are rich enough to reproduce the full statistical behaviour of ζ, while the case α=1 (plain log t) is a genuine boundary where the method provably stops. A sympathetic reader would care because these are the slowest shifts for which value-distribution results were still open.

What carries the argument

The argument rests on moment estimates for Dirichlet polynomials evaluated along a shifted curve. For a shift γ, the first derivative test converts decay of ∫ (n/m)^{iγ(t)} dt into powers of 1/γ'(T); the class F' is designed so that this decay is small and the preimage of exceptional zero-neighbourhoods is controlled. For logarithmic shifts γ(t)=(log t)^α, the derivative γ'(t)=α(log t)^{α-1}/t is small, so the paper instead uses a zero-density estimate N(σ,T)≪T^{Φ(σ)+ε} to show that the measure of t for which (log t)^α lands near a zero is o(T), as long as σ_0 > x_Φ(α_1); under RH the exceptional set vanishes unconditionally. The Fourier convergence for powers with α>1 is what separates the

What would settle it

Find one rectangle R and one sequence T_k for which |P_{T_k}(log ζ(σ+i(log t)^r)∈R)-P(log ζ(σ,X)∈R)| grows faster than ((r-1) log log T_k)^{-σ}, with r>1 and σ>x_Φ(r); that would disprove the main quantitative theorem.

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Extended reading notes

Core claim

For 1<α_1<...<α_r, the vector (log ζ(s+i(log τ)^{α_1}),...,log ζ(s+i(log τ)^{α_r})) is jointly universal in the strip x_Φ(α_1)<σ<1, and under the Riemann hypothesis in the whole strip 1/2<σ<1. In the same range, the discrepancy between the empirical distribution of log ζ(σ+i(log t)^r) and the random Euler product limit is O(((r-1) log log T)^{-σ}). The paper also proves a unified discrepancy estimate D_{σ,γ}(T)≪(log γ(T))^{-σ} for every shift in the class F'. The border case γ(t)=log t is excluded: the key Fourier integral does not converge, and the method forces the Dirichlet polynomial length to be constant.

Load-bearing premise

The load-bearing premise is that the relevant shift has exponent α>1, so that the derivative of (log t)^α eventually dominates the reciprocal of t; if the shift is plain log t, the key average fails to converge and the proof collapses.

Editorial extensions

If this is right

  • The Bohr-Jessen limit theorem holds with a quantitative rate for every shift in F', including e^t, t^a, and iterated exponentials, with the rate depending only on the size of log γ(T).
  • Universality under logarithmic-power shifts holds jointly for any finite number of distinct exponents α>1, so the shift can be arbitrarily slow as long as it is faster than log t.
  • Under RH, both universality and discrepancy estimates extend to the full strip 1/2<σ<1, matching the classical picture.
  • The α=1 case is shown to be a methodological barrier: the same Fourier-averaging approach cannot even produce the limit distribution for log t.
  • The discrepancy bound D_{σ,r}(T) ≪ ((r-1) log log T)^{-σ} degrades as r→1, quantifying exactly why r=1 is hard.

Reading between the lines

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

  • The method suggests a critical threshold at α=1: logarithmic shifts at or below log t likely require a different probabilistic model, perhaps with an averaging kernel that handles persistent phases, rather than the standard first-derivative test.
  • If the Fourier-averaging obstruction is structural, then for γ(t)=log t even the existence of a limiting distribution may fail or require Cesàro-type weights; checking the limit with such weights would be a quick test.
  • The zero-density threshold x_Φ(α_1) is not optimal in the unconditional statement; any future improvement of zero-density estimates directly widens the universality strip.
  • The joint universality result likely extends to other L-functions and to shifts of the form c (log t)^α with distinct constants, as long as linear independence of log p is preserved.
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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 / 5 minor

Summary. This paper studies the value distribution of the Riemann zeta-function under general shifts. Theorem 1.4 gives a discrepancy bound D_{σ,γ}(T) ≪ (log γ(T))^{-σ} for shifts in a class F′. Theorem 1.5 proves joint universality for logarithmic-power shifts (log τ)^{α_j}, 1<α_1<...<α_r, in a strip depending on the zero-density exponent Φ, and under the Riemann hypothesis extends the strip to 1/2<σ<1. Theorem 1.6 gives a quantitative Bohr–Jessen discrepancy D_{σ,r}(T) ≪ ((r−1) log log T)^{-σ} for the single shift (log t)^r. The proofs combine zero-density estimates, Dirichlet polynomial approximations, Beurling–Selberg smoothing, and the probabilistic Bagchi–Matsumoto framework. The main weaknesses are a false estimate in Lemma 6.5 and the largely deferred proofs of the key statements in Section 7.2.

Significance. If the results are correct, the paper makes a genuine contribution: it unifies discrepancy estimates for a broad class of shifts (Theorem 1.4) and, more strikingly, establishes joint universality and quantitative value-distribution results for logarithmic-power shifts, which are outside the polynomial/exponential coverage of previous work. The methods are mostly standard and the use of zero-density estimates is principled; there are no fitted parameters. The paper ships no code or machine-checked proofs, but the probabilistic framework is reproducible from the references. However, the validity of the central universality claim currently hinges on a false estimate in Lemma 6.5, and the main quantitative theorem (Theorem 1.6) rests on unproved 'repetition' arguments. These are local and repairable, but they must be fixed before the results can be accepted.

major comments (3)
  1. [§6, Lemma 6.5] The displayed estimate '≪(log log T)^{-1}' in the proof of Lemma 6.5 is false for the chosen Y=(log log T)^{4/(σ0−1/2)}. Applying Lemma 4.1 with the endpoint of the zeta interval T_ζ=(log T)^{α_j} gives a pointwise error O(Y^{(1/2−σ0)/2}(log T_ζ)^3)=O((log log T)^{-2}(log log T)^3)=O(log log T), not o(1). Consequently the L^1 approximation of logζ by logζ_X on X_K(T) is not established, and Proposition 6.1, hence Theorem 1.5, lacks a valid proof. The gap is concrete and fixable: taking Y=(log log T)^N with N>8/(σ0−1/2) makes the first error tend to 0 while the errors in Lemmas 6.2 and 6.3 remain o(1).
  2. [§7.2, Lemma 7.2 and Proposition 7.3] The proof of Theorem 1.6 rests on Lemma 7.2 and Proposition 7.3, but neither is actually proved. Lemma 7.2 is dismissed with 'we can prove this lemma in the same way as Lemma 4.3', and Proposition 7.3 with 'the proposition follows by repeating the proof of Proposition 4.5'. These are the decisive estimates that produce the (r−1) log log T discrepancy. Because γ(t)=(log t)^r is not in the class F (it fails (F3)), the repetition is not completely formal; the parameter choices and error terms must be checked. The authors should supply the full arguments, in particular the verification that the analogue of (4.2) holds with error O((log log T)^{-5}).
  3. [§7.2, proof of Theorem 1.6] The proof sets L=log log log T at the start, but then concludes D_{σ,r,Y}(T)≪(log log T)^{-2}+L^{-1}≪((r−1)log log T)^{-σ}. For L=log log log T, L^{-1}=1/log log log T, which is not O((log log T)^{-σ}) for any σ>0. The final inequality only follows if L is the quantity c_3((r−1)log log T)^σ from Proposition 7.3. As printed, the derivation of the stated discrepancy bound is not justified. Please correct the definition of L and the associated display.
minor comments (5)
  1. [§4, Lemma 4.2] The final exponent '1/2−σ0' in Lemma 4.2 does not follow from the cited zero-density estimate [28, Theorem 9.19.A] and the preceding display; for σ0>1/2 it would imply N(σ0,T) decays, which is impossible. The exponent appears to be a typo; please correct it to the value actually obtained from the zero-density theorem and condition (F4).
  2. [Notation] The symbol r is used both as the number of joint components in Theorem 1.5 and as the exponent in Theorem 1.6. This is confusing, especially in Section 7 where both roles occur; please use a different symbol, e.g. ν, for one of them.
  3. [§6, Lemma 6.5] The notation ℓ([...]; σ0, Y) is used in the proof of Lemma 6.5, but Lemma 4.1 defines ℓ only with a single parameter y. Please align the definitions so that the role of σ0 and Y is clear.
  4. [§4, Proposition 4.5] The proof states 'Note that we use (F4) to evaluate ...' but the verification is not shown. Since condition (F4) is a new structural assumption, a few lines explaining the estimate would be helpful.
  5. [Abstract/Introduction] The phrase 'logarithmic-power shifts' should specify that α>1 is assumed; the case α=1 is explicitly out of reach and is only discussed in §8.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity; central derivations are self-contained, with one minor non-load-bearing self-citation and a separate Lemma 6.5 correctness gap.

full rationale

The derivation chain is independent of its conclusions. Theorem 1.4's discrepancy bound is proved from Lemma 4.1 (Endo), Beurling–Selberg smoothing, and a Fourier/characteristic-function argument (Proposition 3.1, Lemmas 3.2–3.5, Proposition 4.5); no parameter is fitted to the discrepancy being predicted. Theorem 1.5 and Theorem 1.6 use standard zero-density input N(σ,T)≪T^{Φ(σ)+ε}, Lemma 6.6's first-derivative-test equidistribution for log-power phases, and approximation lemmas; the threshold x_Φ(α1) is chosen by definition to make the exceptional set vanish, which is a construction, not a circular reduction. The only self-citation, [22], supplies the class F definition and context ('the author [22] introduced the class F') but no theorem used as input; it is not load-bearing. The paper openly states its α=1 limitation (Section 8: 'establishing a limit theorem for the logarithmic shift would require some fundamentally different approach'), which is a limitation, not circularity. One substantive correctness concern (not circularity) is in Lemma 6.5: the bound '≪(log log T)^{-1}' for the first term does not follow from Lemma 4.1 as written, because the zeta-argument endpoint is (log T)^{α_j}, making Lemma 4.1's log T equal O(log log T), so the pointwise error is O((log log T)^{-2}(log log T)^3)=O(log log T), not o(1). This is a proof gap in the approximation step for Proposition 6.1, but it involves no fitted input, self-definition, or self-citation chain.

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

No free parameters: all constants are universal and no data are fitted. The results rest on standard external inputs (zero-density estimates, Mergelyan, probabilistic tools) and the paper's own shift-class definitions.

assumptions (5)
  • domain assumption Zero-density estimate N(σ,T) ≪ T^{Φ(σ)+ε} for some non-increasing Φ
    Invoked to define x_Φ(α) and to control exceptional intervals in Lemmas 6.2 and 7.1; specific Φ from Ingham and Guth–Maynard.
  • domain assumption Riemann hypothesis (conditional)
    Used only in the conditional strengthenings of Theorems 1.5 and 1.6 to extend the σ-range to 1/2<σ<1.
  • domain assumption Support of logζ(s,X) in H(R) is the whole space H(R)
    Cited from [5, Proposition 2]; required to infer universality from the weak convergence of Q_T to Q.
  • standard math Mergelyan's theorem, Portmanteau's theorem, first-derivative test
    Standard analytic tools used in Sections 6–7 to approximate functions and evaluate Fourier integrals.
  • ad hoc to paper Class F′ conditions (F1)–(F4)
    Defines the scope of Theorem 1.4; the definition is new to this paper, though natural. The theorem claims nothing outside this class.

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Pith. "Pith review of On the value distribution of the Riemann zeta-function with general shifts." pith.science (2026). https://pith.science/paper/GGT5HH2A

@misc{pith2026260720041,
  author       = {Pith},
  title        = {Pith review of: On the value distribution of the Riemann zeta-function with general shifts},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GGT5HH2A}},
  note         = {Machine review of arXiv:2607.20041}
}
read the original abstract

This paper studies the value distribution of the Riemann zeta-function under general shifts. We establish discrepancy estimates for the Bohr--Jessen limit theorem for a broad class of shifts. We also prove a universality theorem for logarithmic-power shifts, which are not covered by previous universality results for general shifts. Furthermore, we obtain a quantitative Bohr--Jessen limit theorem for logarithmic-power shifts.

Discussion (0). Continue with ORCID to comment.

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