THE FREQUENCY AND THE STRUCTURE OF LARGE CHARACTER SUMS

Abstract. Let M() denote the maximum of j∑nN (n)j for a given non-principal Dirichlet character (mod q), and let N denote a point at which the maximum is at-tained. In this article we study the distribution of M()= p q as one varies over characters (mod q), where q is prime, and investigate the location of N. We show that the dis-tribution of M()= p q converges weakly to a universal distribution , uniformly through-out most of the possible range, and get (doubly exponential decay) estimates for 's tail. Almost all for which M() is large are odd characters that are 1-pretentious. Now, M() j∑nq=2 (n)j = j2(2)j pqjL(1; )j, and one knows how often the latter expres-sion is large, which has been how earlier lower bounds on were mostly proved. We show, though, that for most with M() large, N is bounded away from q=2, and the value of M() is little bit larger tha