3
$\begingroup$

Several formulations of the Riemann hypothesis can be given in terms of Euler’s constant. For example,

Theorem (Nicolas 1981) The Riemann hypothesis holds if and only if all the primorial numbers $N_k = p_1p_2 · · · p_k$ with $k ≥ 2$ satisfy $$\dfrac{N_k}{\phi(N_k) \ln \ln N_k} > e^{\gamma}.$$

or

Theorem (Nicolas 2012) For each integer $n ≥ 2$, the Riemann hypothesis is equivalent to the statement that $$\lim_{n \to \infty} \sup (\dfrac{n}{\phi(n)} - e^{\gamma} \ln \ln n ) \sqrt{\ln n} = e^γ (4 + γ − \ln (4π)) $$

After searching quite a lot on internet, it seems that only constant itself appears in connections with the Riemann hypothesis. Are there any theorem at all relating transcendency/irrationality of $\gamma$ and the Riemann hypothesis?

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy