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Assuming the Riemann hypothesis, this paper proves explicit bounds for ζ and its logarithmic derivative on Re s = 1 with uniform error terms valid for all t ≥ e^18, improving the lower-order error from (log log t)^2/log t to log log t/log t

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Under RH, the paper derives new explicit bounds for Re(ζ'/ζ), |ζ|, |1/ζ|, and |ζ'/ζ| on Re(s)=1, improving known lower-order constants.

T0 review reviewed 2026-08-03 challenge →

load-bearing objection Solid conditional improvement; the advertised constants depend on unverified numerical inequalities that a referee should pin down. the 3 major comments →

arxiv 2602.06199 v2 pith:DKUCYVST submitted 2026-02-05 math.NT

Explicit conditional bounds for $\zeta(s)$ at the edge of the critical strip

Andr\'es Chirre , Blas Molero Ravines This is my paper
classification math.NT MSC 11M0611M2641A30
keywords Riemann zeta-functionRiemann hypothesisbandlimited majorantslogarithmic derivativeexplicit boundscritical stripPoisson kernelextremal functions
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The reading

Assuming the Riemann hypothesis, this paper gives explicit, uniform bounds for ζ and its logarithmic derivative on the line Re s = 1, valid for every t ≥ e^18 (about 6.57 × 10^7). The central improvement is in the lower-order terms: |ζ'/ζ(1+it)| is shown to be ≤ 2 log log t − 0.4989 + 9.0581 log log t/log t − 4.7/log t, replacing the previous (log log t)^2/log t error by a term one log-factor smaller, and in particular ≤ 2 log log t for t ≥ 10^30. For |ζ(1+it)| and 1/|ζ(1+it)|, the paper refines the century-old asymptotic estimates by improving the constants inside the 1/log log t corrections and introducing negative lower-order terms. All bounds are conditional: if the Riemann hypothesis fails, they do not apply.

Core claim

The paper establishes explicit two-sided bounds for Re ζ'/ζ(1+it) under the Riemann hypothesis, for all t ≥ e^18. As consequences it obtains: |ζ'/ζ(1+it)| ≤ 2 log log t + 0.0784 − γ + 9.0581 log log t/log t − 4.7/log t (so that in particular |ζ'/ζ(1+it)| ≤ 2 log log t for t ≥ 10^30); |ζ(1+it)| ≤ 2e^γ( log log t − log 2 + 1/2 + 0.2674/log log t − 2.676 log log t/log t ); and 1/|ζ(1+it)| ≤ (12e^γ/π^2)( log log t − log 2 + 1/2 + 5/(8 log log t) + 10.7084/(log log t)^2 ). These are the sharpest explicit conditional bounds currently known on the 1-line, with optimized constants in all lower-order terms.

What carries the argument

The central mechanism is the Poisson kernel h(x) = 1/(2(1/4 + x^2)), which enters through the identity Re ζ'/ζ(1+it) = Σ_γ h(t−γ) − 1/2 log(t/2π) + O(1/t^2). Because h is not bandlimited, the paper replaces it by extremal bandlimited majorants and minorants h^±_∆: entire functions of exponential type 2π∆ that lie above (resp. below) h and are optimal in the one-sided bandlimited approximation sense, with Fourier transforms supported on [−∆,∆]. Substituting h^±_∆ into the Guinand–Weil explicit formula converts the zero sum into gamma-function, prime-power, and boundary terms; a refined Stirling expansion of the gamma term retains lower-order corrections, and optimizing the bandwidth by taking

Load-bearing premise

The Riemann Hypothesis: all main theorems assume every non-trivial zero of ζ(s) has real part 1/2; if a single non-trivial zero lies off that line, the bounds have no standing without substantial modification.

What would settle it

Compute |ζ'/ζ(1+it)| to rigorous precision at any t ≥ e^18 (for instance t = 10^k for k = 8, 9, ...). If any value exceeds 2 log log t − 0.4989 + 9.0581 log log t/log t − 4.7/log t, Theorem 3 is false. More fundamentally, locating a non-trivial zero off the line Re s = 1/2 would falsify the hypothesis under which all bounds are proved.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • If the theorem is correct, the conditional order of magnitude of ζ'/ζ on the 1-line is settled: for t ≥ 10^30 it is at most 2 log log t, matching the expected asymptotic and improving the previous explicit range.
  • The one-sided bounds on Re ζ'/ζ(1+it) directly imply refined estimates for |ζ(1+it)| and 1/|ζ(1+it)|, valid uniformly for all t ≥ e^18, with smaller 1/log log t constants and new negative lower-order corrections.
  • The error term in the log-derivative bound is reduced by a full log factor: from (log log t)^2/log t to log log t/log t, making the bound numerically meaningful at substantially smaller heights.
  • The optimized choice e^{π∆} = log t / 2 shows the leading 2 log log t is the only thing that sets the scale; the error terms come from the Stirling expansion and prime sums rather than from a rebalancing of the bandwidth.

Where Pith is reading between the lines

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

  • The same bandlimited-majorant-plus-refined-Stirling recipe should transfer to other L-function families, yielding the same one-log-factor reduction in error terms for their logarithmic derivatives on the edge of the strip.
  • The bounds can be stress-tested numerically: a rigorous high-precision evaluation of |ζ'/ζ(1+it)| at some t ≥ e^18 that violates the stated inequality would not refute RH but would point to a specific flaw in the proof, while agreement would support the claimed shape of the error term.
  • The method could be pushed further by keeping even more terms in the Stirling expansion, which would likely lower the constants 9.0581 and 4.7 and extend the range of t for which the bound dips below 2 log log t.
  • A natural next target is the derivative (ζ'/ζ)' on the same line, using the same extremal functions; the refined gamma term developed here would enter directly into its explicit formula.
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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

3 major / 4 minor

Summary. Assuming the Riemann hypothesis, the paper proves explicit two-sided bounds for ±Re ζ'/ζ(1+it) for t ≥ e^18 (Theorem 1), then uses them to refine the Lamzouri–Li–Soundararajan bounds for |ζ(1+it)| and |1/ζ(1+it)| (Theorem 2) and to improve the Chirre–Valås–Simonič bound for |ζ'/ζ(1+it)| (Theorem 3). The main advertised novelty is that the lower-order error term in the bound for |ζ'/ζ(1+it)| is reduced from (log log t)^2/log t to log log t/log t, with an additional negative 1/log t term. The proofs combine the Guinand–Weil explicit formula with extremal bandlimited majorants/minorants of the Poisson kernel, together with Perron-type identities and estimates adapted from [10].

Significance. If all the numerical and calculus checks are made fully rigorous, this is a meaningful contribution to explicit conditional estimates on the edge of the critical strip. The proofs are structurally detailed, the Riemann hypothesis is an explicit and transparent premise rather than a hidden assumption, and the application of the Beurling–Selberg extremal machinery is coherent. The improvement over (1.3) is conceptually interesting and the refinements of (1.1)–(1.2) are useful. However, several load-bearing constants are justified only by statements such as 'numerically one can verify' or 'it can be shown', without reproducible code or complete analytic verification. Since these constants determine the advertised improvements, the paper is not yet in a fully acceptable state.

major comments (3)
  1. [§4.1 / footnote 3] The constants η+ = 8.6544 and η− = 6.9856 in (4.3), and hence Theorems 1–3, depend on the assertion that the functions 2e^{πΔ}ε±(Δ)∓8 log(2e^{πΔ}) are decreasing for πΔ ≥ log 9. This is stated only in the footnote ('It can be shown') with no proof. This is a load-bearing calculus check; please provide a complete proof or certified interval-arithmetic code.
  2. [Lemma 12] The extension of (5.1) from x ≥ 100 to x ≥ 81 is essential for the lower bound (1.5). For p = 2 in the range 81 ≤ x ≤ 100, the argument reduces the problem to a trigonometric polynomial whose minimum is asserted to be 0 'by numerical verification', with no code or rigorous error bound. Similarly, the later claim that the supremum over 81 ≤ x ≤ (73.2)^2 of the displayed expression is at most 0.249 is stated only as 'Numerically'. Since the constant 0.249 appears in the final bound (1.5), these finite verifications must be made reproducible and rigorous.
  3. [§7, after (7.6)] The final absorption step leading to Theorem 3, namely the assertion that the combination of error terms is < 3.648/log t for t ≥ e^18, is stated without derivation. This step is load-bearing because the advertised improvement over (1.3) depends on the resulting coefficient 9.0581 and the negative 4.7/log t term. Please provide the explicit estimate, including how the 13652/log^6 t term, the 17.308/log^2 t term, and the 3.2/t^2 term are dominated by 3.648/log t in the stated range.
minor comments (4)
  1. [Theorem 2 / §6.1] The theorem states the coefficient of log log t/log t as 2.6, while the proof of (1.4) concludes with 2.676. Since 2.6 gives a weaker bound than 2.676, the theorem is valid but the mismatch is confusing; please harmonize the display and the proof.
  2. [Theorem 3 / §7] The theorem states −4.7/log t, while the proof obtains −4.773/log t before 'This implies the desired result'. Again, the stated constant is weaker, but the discrepancy should be noted or the stronger constant inserted.
  3. [Lemma 8] The bound '2(e^{πΔ}+e^{-πΔ})/c±_Δ ≤ 3.325...' is asserted without derivation. This is a minor issue, but since the paper aims at explicit constants, a one-line verification would be helpful.
  4. [General] The paper uses 'numerically one can verify' in several places without indicating whether the verification is fully rigorous or merely floating-point. For an explicit-constants paper, the authors should either supply a proof, include reproducible code, or clearly mark the status of each numerical check.

Circularity Check

0 steps flagged

No significant circularity: the proof derives Theorems 1–3 from the Guinand–Weil explicit formula, prior independent lemmas, and explicit parameter optimization; no fitted quantity is renamed as a prediction.

full rationale

The derivation chain is internally coherent and non-circular. Theorem 1 is proved directly from Lemma 4 (partial fraction plus Stirling), Lemma 5 (Guinand–Weil), and the explicit bandlimited majorants/minorants taken from [2], with each term estimated in Lemmas 6–8. Theorems 2 and 3 use Theorem 1 as an input, together with standard lemmas from Lamzouri–Li–Soundararajan [10] and Selberg's moment formula; they do not assume the conclusions they prove. The constants are obtained by optimizing explicit expressions (e.g., c0 = 1.0467 in §6.1, λ0 = 2.1862 in §7), not by fitting to the target inequalities. The Riemann hypothesis is an explicit external assumption, not a conclusion of the paper. The self-citations to [2], [6], and [7] are published results with stated assumptions that do not include the present target bounds; they supply tools (bandlimited functions, known conditional estimates) rather than the final bounds, so they do not create circularity. The paper does contain verification gaps that are correctness/computation risks rather than circularity: footnote 3 asserts without proof that certain functions are decreasing; Lemma 12 relies on a numerical check that a trigonometric polynomial has minimum 0; §7 absorbs several terms into the bound '< 3.648/log t' without derivation. These could affect the constants if wrong, but they are not steps where a claimed result is equivalent to an input by construction.

Axiom & Free-Parameter Ledger

3 free parameters · 2 axioms · 0 invented entities

The paper introduces no new mathematical objects. The 'free parameters' are optimization parameters in the proof, not degrees of freedom fitted to the target bounds. The Riemann hypothesis is the main external assumption.

free parameters (3)
  • Optimization parameter c in Theorem 2 = 1.0467
    Chosen to minimize an expression involving 2/c + A(c)^2; it is an optimization parameter, not fitted to data.
  • Optimization parameter λ in Theorem 3 = 2.1862
    Chosen to minimize the constant term in the bound for |ζ'/ζ|.
  • Constants η+ and η− = 8.6544 and 6.9856
    Numerically obtained upper bounds for certain ε±(Δ) terms; these are proven by monotonicity arguments, not fitted.
axioms (2)
  • domain assumption Riemann Hypothesis
    All main theorems assume RH; the proof of the zero sum estimates and the use of the Guinand–Weil formula rely on it.
  • standard math Standard analytic number theory estimates (Stirling, Perron, ψ(u) bounds)
    The paper invokes known results, e.g., Stirling's formula, Perron's formula, and Schoenfeld's bound for ψ(u), from cited references.

reviewed 2026-08-03 · how reviews work

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Cite this review

Pith. "Pith review of Explicit conditional bounds for $\zeta(s)$ at the edge of the critical strip." pith.science (2026). https://pith.science/paper/DKUCYVST

@misc{pith2026260206199,
  author       = {Pith},
  title        = {Pith review of: Explicit conditional bounds for $\zeta(s)$ at the edge of the critical strip},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DKUCYVST}},
  note         = {Machine review of arXiv:2602.06199}
}
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abstract

In this paper, we obtain explicit bounds for the real part of the logarithmic derivative of the Riemann zeta-function on the line $\re s=1$, assuming the Riemann hypothesis. The proof combines the Guinand--Weil explicit formula with extremal bandlimited majorants and minorants for the Poisson kernel. As an application, we revisit the classical estimates of Littlewood for the modulus of the Riemann zeta-function and of its reciprocal on the line $\re{s}=1$, and derive a slight refinement of the bounds of Lamzouri, Li, and Soundararajan. In addition, we establish an explicit bound for the modulus of the logarithmic derivative of the Riemann zeta-function on the line $\re{s}=1$ under the Riemann hypothesis, improving the lower-order term in a result of Chirre, Val{\aa}s, and Simoni\v{c}.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

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Reference graph

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This paper was first reviewed by deepseek-v4-flash on August 3, 2026.