A note on primes in certain residue classes

Given positive integers $a_1,\dots ,a_k$, we prove that the set of primes $p$ such that $p\equiv ̸1a_i$ for $i=1,\dots ,k$ admits asymptotic density relative to the set of all primes which is at least $\prod_{i=1}^k \left(1-\frac{1}{\varphi(a_i)}\right)$,where$\varphi$ is the Euler’s totient function. This result is similar to the one of Heilbronn and Rohrbach, which says that the set of positive integer $n$ such that $n\equiv ̸0a_i$ for $i=1,\dots ,k$ admits asymptotic density which is at least $\prod_{i=1}^k \left(1-\frac{1}{a_i}\right)$.