  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.
