Nicolas Inequality & RH Criterion

• Nicolas Inequality is an inequality involving Euler’s totient function at primorial arguments that becomes equivalent to the Riemann Hypothesis when holding for all n.
• It leverages asymptotic estimates from Mertens’ theorem and Chebyshev’s function, translating prime product behavior into iterated logarithmic conditions.
• The criterion reveals challenges such as non-monotonicity and oscillatory prime gaps, underscoring deep connections between multiplicative functions and RH.

The Nicolas inequality is an inequality at primorial arguments of Euler’s totient function. In the formulation used in several arXiv treatments, if $p_1,p_2,\dots$ are the consecutive primes and $N_n=\prod_{k=1}^n p_k$, then the inequality is

$$e^\gamma \log\log N_n \;<\; \frac{N_n}{\varphi(N_n)},$$

equivalently,

$$\varphi(N_n)\;>\; e^{-\gamma}\,\frac{N_n}{\log\log N_n},$$

where $\gamma$ is the Euler–Mascheroni constant. Nicolas’s 1983 theorem, as summarized in later preprints, identifies this inequality as a criterion for the Riemann Hypothesis: its validity for every $n\ge 1$ is equivalent to RH, whereas if RH fails then the inequality holds for infinitely many $n$ and fails for infinitely many $n$ (Choie et al., 2010).

1. Canonical formulation and notation

Let $p_n$ denote the $n$-th prime and let the $N_n=\prod_{k=1}^n p_k$0-th primorial be

$N_n=\prod_{k=1}^n p_k$1

For the variant indexed by a real parameter $N_n=\prod_{k=1}^n p_k$2, one also writes

$N_n=\prod_{k=1}^n p_k$3

Euler’s totient function satisfies

$N_n=\prod_{k=1}^n p_k$4

so the Nicolas inequality may be written either in totient form or in reciprocal form. The Chebyshev function

$N_n=\prod_{k=1}^n p_k$5

is a natural auxiliary object because primorials are logarithmically encoded by prime sums (Galdames-Bravo, 14 Sep 2025).

Several preprints use the shorthand $N_n=\prod_{k=1}^n p_k$6 and $N_n=\prod_{k=1}^n p_k$7 when discussing the criterion asymptotically (Choie et al., 2010). In that notation, the inequality takes the compact form

$N_n=\prod_{k=1}^n p_k$8

A notational issue appears in one 2018 exposition: it states both $N_n=\prod_{k=1}^n p_k$9 and, as an “equivalent” reformulation, $$e^\gamma \log\log N_n \;<\; \frac{N_n}{\varphi(N_n)},$$0 (Milner-Gulland, 2018). These two displayed inequalities point in opposite directions. The formulations in (Choie et al., 2010) and (Galdames-Bravo, 14 Sep 2025) use the direction

$$e^\gamma \log\log N_n \;<\; \frac{N_n}{\varphi(N_n)},$$1

which is also the form adopted in the 2025 preprint “On the verification of a Nicolas inequality” (Galdames-Bravo, 14 Sep 2025).

2. Relation to the Riemann Hypothesis

The central significance of the Nicolas inequality lies in Nicolas’s 1983 equivalence theorem. As reported in the later literature, one has the following dichotomy. If RH holds, then for every $$e^\gamma \log\log N_n \;<\; \frac{N_n}{\varphi(N_n)},$$2,

$$e^\gamma \log\log N_n \;<\; \frac{N_n}{\varphi(N_n)},$$3

Conversely, if RH fails, then the same inequality holds for infinitely many $$e^\gamma \log\log N_n \;<\; \frac{N_n}{\varphi(N_n)},$$4 but also fails for infinitely many $$e^\gamma \log\log N_n \;<\; \frac{N_n}{\varphi(N_n)},$$5 (Choie et al., 2010).

A closely related formulation appears in terms of the sequence $$e^\gamma \log\log N_n \;<\; \frac{N_n}{\varphi(N_n)},$$6 of products of the first $$e^\gamma \log\log N_n \;<\; \frac{N_n}{\varphi(N_n)},$$7 primes. The 2025 paper records Nicolas’s criterion as the alternative between

• $$e^\gamma \log\log N_n \;<\; \frac{N_n}{\varphi(N_n)},$$8: $$e^\gamma \log\log N_n \;<\; \frac{N_n}{\varphi(N_n)},$$9 for every $$\varphi(N_n)\;>\; e^{-\gamma}\,\frac{N_n}{\log\log N_n},$$0,
• $$\varphi(N_n)\;>\; e^{-\gamma}\,\frac{N_n}{\log\log N_n},$$1: $$\varphi(N_n)\;>\; e^{-\gamma}\,\frac{N_n}{\log\log N_n},$$2 infinitely often,

together with the equivalence

$$\varphi(N_n)\;>\; e^{-\gamma}\,\frac{N_n}{\log\log N_n},$$3

(Galdames-Bravo, 14 Sep 2025).

This criterion is structurally notable because it converts RH into an inequality involving only primorials, the totient function, and iterated logarithms. A plausible implication is that the criterion appears elementary only at the surface level: the proofs and reformulations consistently reintroduce deep information about the distribution of primes, especially through Mertens-type estimates and oscillations of Chebyshev’s function.

3. Asymptotic background and equivalent reformulations

The asymptotic motivation for the Nicolas inequality is Mertens’ theorem. In the form used by the arXiv discussions,

$$\varphi(N_n)\;>\; e^{-\gamma}\,\frac{N_n}{\log\log N_n},$$4

and, after replacing $$\varphi(N_n)\;>\; e^{-\gamma}\,\frac{N_n}{\log\log N_n},$$5 by $$\varphi(N_n)\;>\; e^{-\gamma}\,\frac{N_n}{\log\log N_n},$$6, one obtains

$$\varphi(N_n)\;>\; e^{-\gamma}\,\frac{N_n}{\log\log N_n},$$7

as $$\varphi(N_n)\;>\; e^{-\gamma}\,\frac{N_n}{\log\log N_n},$$8 (Milner-Gulland, 2018). Thus the Nicolas inequality concerns the sign of a subtle second-order deviation around an asymptotic equality.

The 2025 paper develops a logarithmic reformulation. Writing

$$\varphi(N_n)\;>\; e^{-\gamma}\,\frac{N_n}{\log\log N_n},$$9

it states

$\gamma$0

Combined with the identity

$\gamma$1

where $\gamma$2 and $\gamma$3 is the Mertens constant, and with the estimate

$\gamma$4

the paper rewrites the criterion in terms of the sign of

$\gamma$5

where $\gamma$6 denotes the $\gamma$7 term from Mertens’ formula (Galdames-Bravo, 14 Sep 2025). In this presentation, the eventual behavior of the Nicolas inequality is reduced to a competition between the oscillatory term involving $\gamma$8 and the error term $\gamma$9.

A separate reformulation is announced in “An Equivalent Statement to Nicolas’ Criterion,” which states that Nicolas’s criterion can be reformulated into an equivalent statement and that the reformulation is bounded and monotonic using Chebyshev’s function and results on prime numbers (Bossard, 2015). The available metadata do not include the explicit equivalent statement, but the abstract indicates that the reformulation was designed to clarify how monotonic behavior can and cannot be used in attacking the criterion.

4. Monotonicity strategies and the obstruction from prime gaps

A natural approach is to study the normalized ratio

$n\ge 1$0

at primorial arguments and to define

$n\ge 1$1

By Mertens’ theorem and $n\ge 1$2, one has

$n\ge 1$3

and Nicolas’s inequality is exactly equivalent to

$n\ge 1$4

(Choie et al., 2010).

This suggests an “eventual monotonicity” strategy: if $n\ge 1$5 were strictly decreasing for all sufficiently large $n\ge 1$6, then the limit $n\ge 1$7 would force the entire tail to stay above $n\ge 1$8, yielding Nicolas’s inequality for all large $n\ge 1$9. The same paper translates the stepwise decrease $n$0 into the elementary condition

$n$1

(Choie et al., 2010).

Choie, Planat, and Solé argue that pushing this monotonicity condition for all sufficiently large $n$2 collides with Cramér’s conjecture on prime gaps. Their argument uses Littlewood’s oscillation theorem for $n$3 to produce infinitely many large values where $n$4 exceeds $n$5 by a term of order $n$6, and then shows that eventual strict decrease of $n$7 would require prime gaps much larger than those predicted by

$n$8

(Choie et al., 2010). Under the Cramér-gap assumption, they conclude that $n$9 fails infinitely often, and they also state that the reverse inequality $n$0 likewise fails infinitely often. In this sense, the primorial sequence associated with Nicolas’s criterion cannot be expected to settle into monotonicity of either sign.

The same paper reports an analogous obstruction when Euler’s totient function is replaced by the Dedekind $n$1-function. This parallel underscores that the difficulty is not specific to one multiplicative function but is tied to oscillatory prime-distribution phenomena.

5. Later arXiv treatments and claimed resolutions

The arXiv record contains several preprints that present Nicolas’s inequality not merely as a criterion but as a route to proving or disproving RH. Their claims vary substantially in scope and method.

arXiv id	Reported contribution	Technical focus
(Bossard, 2015)	Equivalent statement to Nicolas’ criterion	Chebyshev-function reformulation; bounded and monotonic behavior
(Milner-Gulland, 2018)	Claims a proof that the Nicolas inequality holds for all $n$2	Auxiliary sequence $n$3, error-parameter $n$4, induction, Dusart bound
(Oliva, 2018)	Claims a disproof of the inequality by contradiction	Product $n$5 rewritten as an alternating sum
(Galdames-Bravo, 14 Sep 2025)	Studies verification through Mertens-error analysis	Sign of $n$6, oscillation of $n$7, conditional implications

The 2018 preprint “A Proof of the Riemann Hypothesis Through the Nicolas Inequality” introduces an auxiliary real sequence $n$8 with $n$9, fixes the threshold $p_n$0, and defines $p_n$1 for $p_n$2 by

$p_n$3

Using Dusart’s bound that for $p_n$4,

$p_n$5

it derives $p_n$6. The paper then defines an “error-parameter” $p_n$7, proves a recurrence for $p_n$8, states a step-size bound

$p_n$9

and claims that $n$0 is strictly increasing for all $n$1, with a direct numerical check $n$2. The conclusion reported there is $n$3 for all $n$4, hence the Nicolas inequality for every $n$5, and therefore RH by Nicolas’s equivalence (Milner-Gulland, 2018).

The 2018 preprint “A disproof of the Riemann Hypothesis via the Nicolas Criterion” claims a contradictory outcome. Its abstract states that it establishes properties of two related sequences, rewrites

$n$6

as an alternating sum, and then derives a contradiction from the assumption that the Nicolas inequality is always true (Oliva, 2018). The available metadata do not provide the detailed argument.

The 2025 paper “On the verification of a Nicolas inequality” isolates the sign of the Mertens error term $n$7 as decisive. It proves the conditional implications:

• if $n$8 for some $n$9 and all $N_n=\prod_{k=1}^n p_k$00, then $N_n=\prod_{k=1}^n p_k$01 holds;
• if $N_n=\prod_{k=1}^n p_k$02 for some $N_n=\prod_{k=1}^n p_k$03 and all $N_n=\prod_{k=1}^n p_k$04, then $N_n=\prod_{k=1}^n p_k$05 holds

(Galdames-Bravo, 14 Sep 2025).

That paper further claims explicit asymptotic bounds

$N_n=\prod_{k=1}^n p_k$06

for all sufficiently large $N_n=\prod_{k=1}^n p_k$07, and from this deduces infinitely many violations of the Nicolas inequality (Galdames-Bravo, 14 Sep 2025). However, the same source also states that the crucial gap is that the true error term in Mertens’s theorem is not known to have one fixed sign for all large $N_n=\prod_{k=1}^n p_k$08, and indeed is believed to oscillate. Accordingly, the paper presents the sign of $N_n=\prod_{k=1}^n p_k$09 as the unresolved linchpin rather than as a fully settled matter.

6. Mathematical significance and related criteria

The Nicolas inequality occupies a characteristic position among “inequality criteria” for RH. It couples a multiplicative function, $N_n=\prod_{k=1}^n p_k$10, to primorial arguments, where prime structure is maximally explicit, and then asks for a global sign assertion just beyond first-order Mertens asymptotics. This framework explains why apparently elementary formulations rapidly lead back to deep analytic inputs such as Mertens estimates, Chebyshev-function oscillation, and conjectural prime-gap bounds (Choie et al., 2010).

The criterion also interacts naturally with other multiplicative-function inequalities. The 2025 verification paper notes the broader interplay between prime-sum error terms and inequalities involving $N_n=\prod_{k=1}^n p_k$11 and $N_n=\prod_{k=1}^n p_k$12, including Robin-type phenomena (Galdames-Bravo, 14 Sep 2025). Within the supplied literature, the closest direct analogue is the Dedekind $N_n=\prod_{k=1}^n p_k$13-function version discussed by Choie, Planat, and Solé: defining

$N_n=\prod_{k=1}^n p_k$14

they consider

$N_n=\prod_{k=1}^n p_k$15

report that RH is equivalent to $N_n=\prod_{k=1}^n p_k$16 for all $N_n=\prod_{k=1}^n p_k$17, and show that the same monotonicity strategy is obstructed by the same Littlewood-oscillation versus Cramér-gap mechanism (Choie et al., 2010).

A persistent misconception is that Nicolas’s criterion is easy because it is stated with elementary functions. The available arXiv treatments point in the opposite direction. Reformulations quickly become sensitive to the behavior of

$N_n=\prod_{k=1}^n p_k$18

to the sign of the $N_n=\prod_{k=1}^n p_k$19 term in the prime reciprocal sum, or to structural properties of sequences such as $N_n=\prod_{k=1}^n p_k$20. This suggests that the Nicolas inequality is best understood not as a simplification of RH in any computationally trivial sense, but as a sharp boundary statement at the interface of multiplicative function theory and fine prime-distribution asymptotics.