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

Abstract: Let $a$ and $q$ be two coprime positive integers. In 1944, Linnik proved his celebrated theorem concerning the size of the smallest prime $p(q,a)$ in the arithmetic progression $a(\mod q)$. In his attempt to prove his result, Linnik established an estimate for the sums of the von Mangoldt function $\Lambda$ on arithmetic progressions. His work on $p(q,a)$ was later simplified, but the simplified proofs relied in one form or another on the same advanced tools that Linnik originally used. The last two decades, some more elementary approaches for Linnik's theorem have appeared. Nonetheless, none of them furnishes an estimate of the same quantitative strength as the one that Linnik obtained for $\Lambda$. In this talk, we will see how one can seal this gap and recover Linnik’s estimate by largely elementary means. The ideas that I will describe build on methods from the treatment of Koukoulopoulos on multiplicative functions with small partial sums and his pretentious proof for the prime number theorem in arithmetic progressions.

Mathematics

Audience: researchers in the topic

Greek Algebra & Number Theory Seminar