In the present paper, we prove that self-approximation of $\log \zeta (s)$
with $d=0$ is equivalent to the Riemann Hypothesis. Next, we show
self-approximation of $\log \zeta (s)$ with respect to all nonzero real numbers
$d$. Moreover, we partially filled a gap existing in "The strong recurrence for
non-zero rational parameters" and prove self-approximation of $\zeta(s)$ for $0
\ne d=a/b$ with $|a-b|\ne 1$ and $\gcd(a,b)=1$.