$Z(t)= \sum_{n=1}^{\lfloor\sqrt t/2\pi\rfloor }\frac{\cos(N(t)-t\log p_{n})}{\sqrt n}$
here $N(t)$ is the smooth part of the zeros and $ p_{n} $ are the primes since $ p_{n} =n\log n $ then $ \log p_{n}=\log n \log\log n $ for big $n$ so the convergence should be pretty similar
but is this approximaiton valid , here the sum over $n$ is made so $ p_{n} \le \lfloor\sqrt t/2\pi\rfloor$