Prescribin the binary digits of the primes, II

We obtain the expected asymptotic formula for the number of primes $p<N=2^n$ with $r$ prescribed (arbitrarly placed) binary digits, provided $r<cn$ for a suitable constant $c>0$. This result improves on our earlier result where $r$ was assumed to satisfy $r<c(\frac n{\log n})^{4/7}$.

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