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arxiv: 1907.07664 · v1 · pith:LLBV7PQSnew · submitted 2019-07-17 · 🧮 math.NT

The Landau function and the Riemann Hypothesis

Marc Deleglise , Jean-Louis Nicolas This is my paper

Pith reviewed 2026-05-24 20:05 UTC · model grok-4.3

classification 🧮 math.NT
keywords Landau functionRiemann hypothesislogarithmic integralsymmetric groupmaximal orderprime powersequivalence
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The pith

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

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

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.

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.

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).

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.

Where Pith is reading between the lines

These are 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.

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.

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.

Figures

Figures reproduced from arXiv: 1907.07664 by Jean-Louis Nicolas, Marc Deleglise.

Figure 1
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. 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 ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

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

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.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 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).

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Reference graph

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