Denote $P(n,k)$ to be the number of primes between $2^n$ and $2^{n+1}-1$ having $k$ number of $1$s in its binary expansion between the $n+1$th binary digit and the least which is always $1$ if $n>1$.

It is clear $\sum_{i=1}^n\sum_{k=0}^{i-1}P(i,k)$ is the number of primes bounded by $2^{n+1}$ and satisfies the square root error bound under Riemann Hypothesis.

Is there a similar error bound at every $i,k$ for $P(i,k)$?