Prescribin the binary digits of the primes, II

TL;DR

This paper derives an asymptotic formula for counting primes less than 2^n with r prescribed binary digits, improving previous bounds and advancing understanding of prime digit patterns.

Contribution

It provides a more precise asymptotic count for primes with prescribed binary digits, extending earlier results to larger r values.

Findings

Established asymptotic formula for primes with prescribed binary digits

Extended the range of r for which the formula holds

Improved bounds from previous work

Abstract

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 $r<c(\frac n{\log n})^{4/7}$.