Prescribing the binary digits of primes

We present a new result on counting primes $p<N=2^n$ for which $r$ (arbitrarily placed) digits in the binary expansion of $p$ are specified. Compared with earlier work of Harman and Katai, the restriction on $r$ is relaxed to $r<c(\frac{n}{\log n}{)}^{4\mathrm{/}7}$. This condition results from the estimates of Gallagher and Iwaniec on zero-free regions of $L$-functions with `powerful’ conductor.