Primes with restricted digits (Q2419717)

For \(a\in \{ 0, \dots , 9\} \), let \(\mathcal{A}_a\) be the set of all positive integers which have no digit \(a\) in their decimal expansions. It is well known that the number of integers in \(\mathcal{A}_a\) which do not exceed \(x\) is \(O(x^{1-\delta }) \) for some positive constant \(\delta \). In this paper, the author proves the following excellent result: There are infinitely many primes in \(\mathcal{A}_a\). In fact, the author proves the following stronger result: \[c_1 \frac{x^{\log 9/\log 10}}{\log x}\le |\{ p\in \mathcal{P} : p\in \mathcal{A}_a, p\le x \} |\le c_2 \frac{x^{\log 9/\log 10}}{\log x} ,\] where \(\mathcal P\) denotes the set of all primes and \(c_1, c_2\) are two positive constants. The proof is based on the circle method, Harman's sieve, the large sieve, the geometry of numbers and so on.