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Let M (χ) denote the maximum of | n≤N χ(n)| for a given non-principal Dirichlet character χ (mod q), and let N χ denote a point at which the maximum is attained. In this article we study the distribution of M (χ)/ √ q as one varies over characters (mod q), where q is prime, and investigate the location of N χ . We show that the distribution of M (χ)/ √ q converges weakly to a universal distribution Φ, uniformly throughout most of the possible range, and get (doubly exponential decay) estimates for Φ's tail. Almost all χ for which M (χ) is large are odd characters that are 1-pretentious. Now, M (χ) ≥ | n≤q/2 χ(n)| = |2−χ(2)| π √ q|L(1, χ)|, and one knows how often the latter expression is large, which has been how earlier lower bounds on Φ were mostly proved. We show, though, that for most χ with M (χ) large, N χ is bounded away from q/2, and the value of M (χ) is little bit larger than √ q π |L(1, χ)|.

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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.

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