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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 $P(i,k)$?

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  • $\begingroup$ The number of 1s in the expansion of a number is generally known as Hamming weight, and while getting that tight of a bound seems unlikely especially at the extremes of $k$, there are in fact things known about them: mathoverflow.net/questions/22629/… $\endgroup$
    – Steven Stadnicki
    May 5, 2021 at 2:48
  • $\begingroup$ @StevenStadnicki I am asking if there is a Riemann Hypothesis for it and not a complete answer. $\endgroup$
    – Turbo
    May 5, 2021 at 3:58

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