A note on primes in certain residue classes

Abstract

Given positive integers $a_1,\dots ,a_k$, we prove that the set of primes $p$ such that $p\not\equiv 1\mathrm{mod}{a}_i$ for $i=1,\dots ,k$ admits asymptotic density relative to the set of all primes which is at least ${\prod }_{i=1}^k\left(1-\frac{1}{\phi (a_i)}\right)$, where $\phi$ 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 $n\not\equiv 0\mathrm{mod}{a}_i$ for $i=1,\dots ,k$ admits asymptotic density which is at least ${\prod }_{i=1}^k\left(1-\frac{1}{a_i}\right)$.

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