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In this work and its sister paper [5] we give a new proof of the famousLinnik theorem bounding the least prime in an arithmetic progression. Usingsieve machinery in both papers, we are able to dipense with the log-free zerodensity bounds and the repulsion property of exceptional zeros, two deepinnovations begun by Linnik and reelied on in earlier proofs.

Sifting for small primes from an arithmetic progression