Mean Values of Character Sums

Abstract

For a non-principal Dirichlet character χ modulo q,Let the Pólya-Vingradov inequality asserts that M(x) < q1/2 log q see [7]. in the opposite direction it is a trivial consequence of lemma 1 below and 1. Parseval's identity that if χ is primitive modulo q, thenWe show that on average the latter of these estimates is the more precise.THEOREM 1. For any real k > 0where the summation is over all non-principal characters modulo q.THEOREM 2. For any k > 0,