  For any natural number $n$, let $X'_n$ be the set of primitive Dirichlet
characters modulo $n$. We show that if the Riemann hypothesis is true, then the
inequality $|X'_{2n_k}|\le C_2 e^{-\gamma} \phi(2n_k)/\log\log(2n_k)$ holds for
all $k\ge 1$, where $n_k$ is the product of the first $k$ primes, $\gamma$ is
the Euler-Mascheroni constant, $C_2$ is the twin prime constant, and $\phi(n)$
is the Euler function. On the other hand, if the Riemann hypothesis is false,
then there are infinitely many $k$ for which the same inequality holds and
infinitely many $k$ for which it fails to hold.
