Pith. sign in

REVIEW 1 major objections 27 references

The Landau function and the Riemann Hypothesis

T0 review · 1 major / 0 minor · reviewed 2026-05-24 · grok-4.3

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read The inequality log g(n) < li^{-1}(n) for all positive integers n holds exactly when the Riemann hypothesis is true.

desk verdict The claimed equivalence cannot hold: the inequality is true for large n by standard asymptotics on g(n) and li^{-1}(n). read the letter →

arxiv 1907.07664 v1 pith:LLBV7PQS submitted 2019-07-17 math.NT

Marc Deleglise , Jean-Louis Nicolas This is my paper · ORCID
classification math.NT
keywords LandaufunctionRiemannhypothesislogarithmicintegralsymmetricgroupmaximalorderprimepowersequivalence
topics The Riemann Hypothesis
open problems 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

This paper proves an equivalence between the Riemann hypothesis and the statement that log g(n) stays strictly below the inverse logarithmic integral li^{-1}(n) for every positive integer n. The Landau function g(n) is the largest order of any permutation of n objects, realized as the biggest product of prime powers whose weighted sum is at most n. The comparison uses the inverse of the logarithmic integral, a function that encodes the prime-number counting behavior central to the zeta function. A reader cares because the result converts one of the most famous unsolved problems into a concrete, in-principle checkable inequality involving a single arithmetic function.

What carries the argument

The Landau function g(n), the largest product of prime powers with sum of the terms at most n, together with its direct comparison against the inverse logarithmic integral li^{-1}(n).

What would settle it

An explicit integer n at which the computed value of log g(n) meets or exceeds li^{-1}(n) would violate the inequality and, by the claimed equivalence, falsify the Riemann hypothesis.

Watch

Extended reading notes

Core claim

The main result is that the property 'For all n > 0, log g(n) < li^{-1}(n)' (where g(n) is the maximal order of an element of the symmetric group of degree n) is equivalent to the Riemann hypothesis.

Load-bearing premise

The known asymptotic and extremal properties of the Landau function g(n), when combined with the definition of li^{-1}, produce an inequality whose validity is exactly coextensive with the Riemann hypothesis.

Editorial extensions

If this is right

  • The Riemann hypothesis holds if and only if the inequality log g(n) < li^{-1}(n) is true for every positive integer n.
  • Any counterexample n to the inequality would immediately disprove the Riemann hypothesis.
  • Asymptotic and extremal results already known for g(n) translate directly into statements about the location of zeta zeros.
  • The growth of the maximal orders in symmetric groups is governed by the same prime-distribution law that appears in the Riemann hypothesis.

Reading between the lines

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

  • Direct computation of g(n) for successively larger n supplies a practical route to test the inequality numerically up to bounds where li^{-1}(n) can still be evaluated accurately.
  • The equivalence supplies a new arithmetic-function formulation that could be compared with other known reformulations of the Riemann hypothesis for possible simplifications or contradictions.
  • If the inequality holds up to a computable limit, it would confirm the hypothesis inside the corresponding range of zeta zeros, though this remains a finite check.
Share X Bluesky LinkedIn Reddit HN

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

1 major / 0 minor

Summary. The manuscript claims that the property 'For all n > 0, log g(n) < li^{-1}(n)' (with g(n) the Landau function) is equivalent to the Riemann hypothesis.

Significance. If the claimed equivalence were valid, it would provide a novel characterization of the Riemann hypothesis in terms of the maximal order of elements in the symmetric group S_n.

major comments (1)
  1. [Abstract] Abstract: the claimed equivalence is incompatible with the known asymptotics log g(n) ∼ √(n log n) (Landau 1909) and li^{-1}(n) ∼ n log n; the inequality therefore holds unconditionally for all sufficiently large n, so the two sides cannot be coextensive.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for identifying a fundamental inconsistency between the claimed equivalence and the known asymptotics of the functions involved. We address the point directly below.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the claimed equivalence is incompatible with the known asymptotics log g(n) ∼ √(n log n) (Landau 1909) and li^{-1}(n) ∼ n log n; the inequality therefore holds unconditionally for all sufficiently large n, so the two sides cannot be coextensive.

    Authors: We agree with the referee. The asymptotic relations log g(n) ∼ √(n log n) and li^{-1}(n) ∼ n log n imply that log g(n) = o(li^{-1}(n)) as n → ∞, so the inequality log g(n) < li^{-1}(n) holds for all sufficiently large n independently of the Riemann hypothesis. The statement that the inequality holds for every n > 0 is therefore equivalent only to a finite verification up to some bound and cannot be coextensive with the Riemann hypothesis. This renders the main claim of the manuscript incorrect as stated. revision: yes

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: equivalence claim stated without self-referential construction or load-bearing self-citation in available text

full rationale

The abstract asserts that ∀n>0 log g(n) < li^{-1}(n) is equivalent to RH, but supplies no derivation chain, equations, or citations that reduce the claimed equivalence to a definition, fit, or prior self-result. No self-definitional steps, fitted inputs renamed as predictions, or ansatz smuggling appear. The derivation is treated as self-contained against external number-theoretic benchmarks (Landau function asymptotics, li inverse) until full text review shows otherwise. This is the default honest finding when no load-bearing reduction is exhibited.

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

The abstract introduces no explicit free parameters, new axioms, or invented entities beyond the standard definitions of g(n) and li^{-1}(n).

how reviews work

0 comments
Cite this review

Pith. "Pith review of The Landau function and the Riemann Hypothesis." pith.science (2026). https://pith.science/paper/LLBV7PQS

@misc{pith2026190707664,
  author       = {Pith},
  title        = {Pith review of: The Landau function and the Riemann Hypothesis},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LLBV7PQS}},
  note         = {Machine review of arXiv:1907.07664}
}
abstract

The Landau function $g(n)$ is the maximal order of an element of the symmetric group of degree $n$; it is also the largest product of powers of primes whose sum is $\le n$. The main result of this article is that the property " For all $n > 0$ , $log g(n) < li^{-1} (n))$ " (where $li^{-1}(n)$ denotes the inverse function of the logarithmic integral) is equivalent to the Riemann hypothesis.

Figures

Figures reproduced from arXiv: 1907.07664 by the authors.

Figure 1
Figure 1. The first superchampion numbers and, for 2 ≤ ≤ 29, = (0) with [PITH_FULL_IMAGE:figures/full_fig_p019_1.png] view at source ↗
Figure 2
Figure 2. Enumeration of super-champion numbers Using a prime generator function, which computes the sequence of successive primes up to in time ( log log ), we wrote a C++ function which computes 1 In [11] the term prefix is used with a different meaning. 36 [PITH_FULL_IMAGE:figures/full_fig_p036_2.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

27 extracted references · 27 canonical work pages

  1. [1]

    Abramowitz and I

    M. Abramowitz and I. A. Stegun. Handbook of Mathematical Functions, Dover Publications, Inc. New-York

    work page

  2. [2]

    C. Axler. New bounds for the sum of the first /u1D45Bprime numbers, ArXiv 1606.06874

    work page Pith review arXiv

  3. [3]

    C. Axler. New estimates for some functions defined over pr imes, Integers, 18 (2018), Paper No A52, 21 pp

    work page 2018

  4. [4]

    Broughan

    K. Broughan. Equivalents of the Riemann Hypothesis, vol . 1. Encyclopedia of Mathematics and its Applications, 164, Cambridge University Press, 2017

    work page 2017

  5. [5]

    J. Büthe. An analytic method for bounding /u1D713 ( /u1D465) . Math. Comp. 87 (2018), no 312, 1991-2009

    work page 2018

  6. [6]

    B.C. Berndt. Ramanujan’s Notebooks, Part IV , 126–131. Springer, New York, 1994

    work page 1994

  7. [7]

    Deléglise and J.-L

    M. Deléglise and J.-L. Nicolas. Le plus grand facteur pre mier de la fonction de Landau, Ramanujan J. 27 (2012), 109–145. 58

    work page 2012

  8. [8]

    Maximal product of primes whose sum is bounded, Proc

    Marc Deléglise and Jean-Louis Nicolas. Maximal product of primes whose sum is bounded, Proc. Steklov Inst. Math. 282 (Issue 1, Supplement) (2013), 73–102

    work page 2013

Show all 27 references
  1. [9]

    On the Largest Pr oduct of Primes with bounded sums, J

    Marc Deléglise and Jean-Louis Nicolas. On the Largest Pr oduct of Primes with bounded sums, J. of Integer Sequences 18, (2013), Article 15.2.8

    2013

  2. [10]

    An arithmetic e quivalence to the Riemann hypothesis, J

    Marc Deléglise and Jean-Louis Nicolas. An arithmetic e quivalence to the Riemann hypothesis, J. Aust. Math. Soc 106 (2019), 235-273

    2019

  3. [11]

    Landau’s func- tion for one million billions, J

    Marc Deléglise, Jean-Louis Nicolas and Paul Zimmerman n. Landau’s func- tion for one million billions, J. Théor. Nombres Bordeaux 20 (2008), 625– 671

    2008

  4. [12]

    P . Dusart. Explicit estimates of some functions over pr imes, Ramanujan J. 45 (2018), 227–251

    2018

  5. [13]

    E. Landau. Über die Maximalordnung der Permutationen g egebenen Grades, Archiv. der Math. und Phys. 5 (1903), 92–103. Handbuch der Lehre von der Verteilung der Primzahlen, I, 2nd ed., Chelsea, 1953, pp. 222–229

    1903

  6. [14]

    Massias and G

    J.-P . Massias and G. Robin. Bornes effectives pour certa ines fonctions con- cernant les nombres premiers, J. de Théorie des Nombres de Bordeaux , 8, 1996, 215–242

    1996

  7. [15]

    Massias, J.-L

    J.-P . Massias, J.-L. Nicolas and G. Robin. Évaluation asymptotique de l’ordre maximum d’un élément du groupe symétrique, Acta Arith. 50 (1988), 221– 242

    1988

  8. [16]

    Majoration explicite de l’ordre maximum d’un élément du groupe symétrique, Ann

    Jean-Pierre Massias. Majoration explicite de l’ordre maximum d’un élément du groupe symétrique, Ann. Fac. Sci. Toulouse Math. 6 (1984), 269–281

    1984

  9. [17]

    Effective bounds for the maximal order of an element in the symmetric group, Math

    Jean-Pierre Massias, Jean-Louis Nicolas and Guy Robin . Effective bounds for the maximal order of an element in the symmetric group, Math. Comp. 53 (188) (1989), 665–678

    1989

  10. [18]

    Miller, The Maximal Order of an Element of a Finite Sym metric Group

    W. Miller, The Maximal Order of an Element of a Finite Sym metric Group. Amer. Math. Monthly, 94, (1987), 497–506

    1987

  11. [19]

    J.-L. Nicolas. Ordre maximum d’un élément du groupe de p ermutations et highly composite numbers, Bull. Soc. Math. France 97 (1969), 129–191

    1969

  12. [20]

    Nicolas, On Landau’s function /u1D454( /u1D45B)

    J.-L. Nicolas, On Landau’s function /u1D454( /u1D45B) . In R. L. Graham, J. Nešetřil and S. Butler, eds., The Mathematics of Paul Erdős II , Springer-Verlag, 2013, pp. 207–220. 59

    2013

  13. [21]

    D. J. Platt and T. Trudgian. On the first sign change of /u1D703( /u1D465) − /u1D465, Math. Comp. 85 (2016), no 299, 1539-1547

    2016

  14. [22]

    Ramanujan

    S. Ramanujan. Highly composite numbers. Proc. London Math. Soc. Serie 2, 14, (1915), 347–409. Collected papers, Cambridge Universi ty Press, 1927, 78–128. Highly composite numbers, annotated and with a fore word by J.-L. Nicolas and G. Robin, Ramanujan J., 1 (1997), 119–153

    1915

  15. [23]

    G. Robin. Permanence de relations de récurrence dans ce rtains développe- ments asymptotiques. Publications de l’Institut Mathématique de Beograd , Nouvelle série, 43 (57), 1988, 17–25

    1988

  16. [24]

    Barkley Rosser and Lowell Schoenfeld, Approximate f ormulas for some functions of prime numbers, Illinois J

    J. Barkley Rosser and Lowell Schoenfeld, Approximate f ormulas for some functions of prime numbers, Illinois J. Math. 6 (1962), 64–94

    1962

  17. [25]

    B. Salvy. Fast computation of some Asymptotic Function al Inverses, J. Sym- bolic Computation, 17, 1994, 227–236

    1994

  18. [26]

    Schoenfeld

    L. Schoenfeld. Sharper bounds for the Chebyshev functi ons /u1D703( /u1D465) and /u1D713 ( /u1D465) . II, Math. Comp., 30, 1976, 337–360

    1976

  19. [27]

    Lyon, Université Claude Bernard Lyon 1, CNRS UMR 5208 Institut Camille Jordan, Mathématiques, Bât

    http://math.univ-lyon1.fr/homes-www/nicolas/calculgdenHR.html Marc Deléglise, Jean-Louis Nicolas, Univ. Lyon, Université Claude Bernard Lyon 1, CNRS UMR 5208 Institut Camille Jordan, Mathématiques, Bât. Doyen Jean Br aconnier, 43 Bd du 11 Novembre 1918, F-69622 Villeurbanne ce...

    1918

Pith tools

Reviewed May 24, 2026 · model on record in the stance chip above.