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

TL;DR

This paper presents a new, uniform estimate for sums of the von Mangoldt function in arithmetic progressions, inspired by Linnik's work and utilizing pretentious number theory techniques.

Contribution

It introduces a pretentious approach to establish a strongly uniform estimate for primes in arithmetic progressions, extending Linnik's classical results with modern methods.

Findings

Established a uniform estimate for the von Mangoldt sums in arithmetic progressions.

Connected pretentious number theory with classical prime distribution results.

Provided insights that could influence future research on 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.