Given positive integers a1, . . . , ak, we prove that the set of primes p such that p = 1 mod ai for i = 1, . . . , k admits asymptotic density relative to the set of all primes which is at least Qk i=1(1 - 1 φ(ai) ), where φ is the Euler 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 = 0 mod ai for i = 1, . . . , k admits asymptotic density which is at least Qk i=1(1 - 1ai).
A note on primes in certain residue classes
LEONETTI, Paolo
;
2018-01-01
Abstract
Given positive integers a1, . . . , ak, we prove that the set of primes p such that p = 1 mod ai for i = 1, . . . , k admits asymptotic density relative to the set of all primes which is at least Qk i=1(1 - 1 φ(ai) ), where φ is the Euler 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 = 0 mod ai for i = 1, . . . , k admits asymptotic density which is at least Qk i=1(1 - 1ai).
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