Prescribing the binary digits of primes - Bourgain, Jean (Israel journal of mathematics, 2013)
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 ( n /log n ) 4/7 . This condition results from the estimates of Gallagher and Iwaniec on zero-free regions of L -functions with ‘powerful’ conductor.