Pretentious multiplicative functions and the prime number theorem for arithmetic progressions

Abstract

Building on the concept of pretentious multiplicative functions, we give a new and largely elementary proof of the best result known on the counting function of primes in arithmetic progressions.

21 pages. Final version. Strengthened Lemma 2.4 to fix an error in the proof of Lemma 4.1. Also made some other minor changes. To appear in Compos. Math

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• Moyennes effectives de fonctions multiplicatives complexes
• A pretentious proof of Linnik's estimate for primes in arithmetic progressions
• The Prime Number Theorem via Karamata's proof of the Landau-Ingham Tauberian theorem
• A general inversion formula for summatory arithmetic functions and its application to the summatory function of the Moebius function
• Shifted moments of quadratic Dirichlet $L$-functions