A pretentious proof of Linnik's estimate for primes in arithmetic progressions

In the present paper, the author adopts a pretentious approach and recovers an estimate obtained by Linnik for the sums of the von Mangoldt function Λ on arithmetic progressions. It is the analogue of an estimate that Linnik established in his attempt to prove his celebrated theorem concerning the size of the smallest prime number of an arithmetic progression. Our work builds on ideas coming from the pretentious large sieve of Granville, Harper, and Soundararajan and it also borrows insights from the treatment of Koukoulopoulos on multiplicative functions with small averages.