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A note on primes in certain residue classes

Abstract

Given positive integers a1,,aka_1,\ldots,a_k, we prove that the set of primes pp such that p≢1modaip \not\equiv 1 \bmod{a_i} for i=1,,ki=1,\ldots,k admits asymptotic density relative to the set of all primes which is at least i=1k(11φ(ai))\prod_{i=1}^k \left(1-\frac{1}{\varphi(a_i)}\right), where φ\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 nn such that n≢0modain \not\equiv 0 \bmod a_i for i=1,,ki=1,\ldots,k admits asymptotic density which is at least i=1k(11ai)\prod_{i=1}^k \left(1-\frac{1}{a_i}\right)

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