Prescribing the binary digits of primes

TL;DR

This paper advances the understanding of prime distribution by counting primes with specified binary digits, relaxing previous restrictions through improved zero-free region estimates of L-functions.

Contribution

It introduces a new bound on the number of specified binary digits in primes, extending prior results with refined analytic number theory techniques.

Findings

Count of primes with r specified binary digits is improved

Restriction on r is relaxed to r< c(n/log n)^{4/7}

Utilizes Gallagher and Iwaniec's zero-free region estimates

Abstract

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 $r< c\Big(\frac n{\log n}\Big)^{4/7}$. This condition results from the estimates of Gallagher and Iwaniec on zero-free regions of $L$-functions with `powerful' conductor.