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A result of Sole, Planat and Omar's paper, ''Quantum mechanics and the Riemann Hypothesis'', (Theorem 2 with $b=2$), says the RH is equivalent to the statement that for every large enough integer $k$, one has

$$\dfrac{\sigma(N_k)}{N_{k}\log\log N_{k}} > \dfrac{6e^{\gamma}}{\pi^2} $$

where $\sigma(u)$ is the sum of divisors function, $\varphi(u)$ the Euler totient function, $\gamma$ the Euler constant and $N_k$ is the product of the first $k$ primes.

Multiplying both sides by $ \dfrac{(N_{K})^2}{\sigma(N_k)\varphi(N_k)}$ gives

$$\dfrac{N_k}{\varphi(N_k)\log\log N_{k}} > \dfrac{6e^{\gamma}(N_{K})^2}{\varphi(N_k)\sigma(N_k) \pi^2}$$

By Nicolas' criterion, the right hand side should be equal to $e^{\gamma}$, which happens only ''at $k=\infty$''.

But for this to make analytic sense, one should consider the left-hand side of the preceding inequality as $k$ approaches infinity, thus the RH is in fact the statement that

as $k$ tends to infinity, the limit of the the left hand side of Nicolas' inequality is greater than or equal to $\gamma$ ?

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  • $\begingroup$ there's a typo in the title $\endgroup$ – YCor Jul 31 '17 at 21:13
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From the paper you mention and the result of Nicolas, the following three statements are known to be logically equivalent:

  1. The Riemann hypothesis.
  2. The inequality $\frac{N_k}{\phi(N_k) \log\log N_k} > e^\gamma$ holding for all $k$. (Nicolas's criterion)
  3. The inequality $\frac{N_k}{\phi(N_k) \log\log N_k} > \frac{6 e^\gamma (N_k)^2}{\varphi(N_k) \sigma(N_k) \pi^2}$ holding for all sufficiently large $k$.

In particular, claim 2 and claim 3 are equivalent. This however does not necessarily imply any relationship between the quantities $e^\gamma$ and $\frac{6 e^\gamma (N_k)^2}{\varphi(N_k) \sigma(N_k) \pi^2}$. This is basically because there is a significant gap between the statistics of arithmetic functions such as $\varphi(N_k)$ and $\sigma(N_k)$ under the assumption of RH, and under the opposite assumption of having a non-trivial zero of zeta off the critical line. This gap allows for the existence of multiple criteria that do not immediately look equivalent to each other, but are nevertheless able to separate the scenarios when RH holds from the scenarios where RH fails.

A more familiar example of this phenomenon arises with the well known equivalences of RH with various bounds on the error term in the prime number theorem. Namely, the following three statements are known to be logically equivalent:

  1. The Riemann hypothesis.
  2. The inequality $|\sum_{n \leq x} \Lambda(n) - x| \leq \frac{1}{8\pi} \sqrt{x} \log^2 x$ holds for all $x \geq 74$. (Schoenfeld's criterion)
  3. For every $\varepsilon > 0$, there exists $C_\varepsilon$ such that the inequality $|\sum_{n \leq x} \Lambda(n) - x| \leq C_\varepsilon x^{1/2+\varepsilon}$ holds for all sufficiently large $x$.

At first glance, the claim 5 is significantly stronger than claim 6, and if the von Mangoldt function $\Lambda$ was replaced by a completely arbitrary function, one could easily cook up examples in which claim 6 was true but claim 5 was false. But the von Mangoldt function is far from being completely arbitrary, being tied to the zeroes of the zeta function by the explicit formula, and the non-trivial zeroes either all lie on the critical line (in which case one can show claim 5), or there is at least one zero off the critical line (in which case one can contradict claim 6 for suitable choices of $x$ and $\varepsilon$).

For an even simpler instance of this sort of gap phenomenon that involves no number theory whatsoever, observe that the following claims are equivalent for any polynomial $P: {\bf R} \to {\bf R}$:

  1. $P$ is constant.
  2. $P$ is bounded.
  3. One has $P(x) = o(|x|)$ as $|x| \to \infty$.

Again, claim 8 appears to be stronger than claim 9 (and claim 7 stronger than claim 8), but they are all equivalent, because there is a significant gap between the behaviour of the constant polynomials and the behaviour of non-constant polynomials.

(But one should mention, though, that it is not difficult to see that the expression $$ \frac{6 e^\gamma (N_k)^2}{\varphi(N_k) \sigma(N_k) \pi^2} = \frac{e^\gamma}{\zeta(2)} \prod_{p \leq p_k} (1 - \frac{1}{p^2})^{-1} = e^\gamma \prod_{p > p_k} (1 - \frac{1}{p^2}) = e^\gamma + O( \frac{1}{k \log k} )$$ does converge reasonably quickly to $e^\gamma$ in the limit $k \to \infty$. So claim 2 and claim 3 are actually not so different from each other, though still not obviously identical. It's more accurate to say that Nicolas's criterion is flexible, rather than ambiguous - one has a little bit of "wiggle room" in that criterion (or in many of the other known criteria for RH), provided by the above-mentioned gap between the RH and non-RH worlds.)

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