Greatest of the least primes in arithmetic progressions having a given modulus

Abstract

We give a heuristic argument, supported by numerical evidence, which suggests that the maximum, taken over the reduced residue classes modulo k , of the least prime in the class, is usually about ϕ ( k ) log ⁡ k log ⁡ ϕ ( k ) \phi (k)\log k\log \phi (k) , where ϕ \phi is Euler’s phi-function.