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

Abstract

In the present paper, we adopt a pretentious approach and prove a strongly uniform estimate for the sums of the von Mangoldt function $\Lambda$ on arithmetic progressions. This estimate is analogous to 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.