We study exponential sums whose coefficients are completely multiplicative and belong to the complex unit disc. Our main result shows that such a sum has substantial cancellation unless the coefficient function is essentially a Dirichlet character. As an application we improve current bounds on odd-order character sums. Furthermore, conditionally on the generalized Riemann hypothesis we obtain a bound for odd-order character sums which is best possible.

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)|.

Let K be a number field containing the n-th roots of unity for some n 3. We prove a uniform subconvexity result for a family of double Dirichlet series built out of central values of Hecke L-functions of n-th order characters of K. The main new ingredient, possibly of independent interest, is a large sieve for n-th order characters. As further applications of this tool, we derive several results concerning L(s, χ) with χ an n-th order Hecke character: an estimate of the second moment on the critical line, a non-vanishing result at the central point, and a zero-density theorem.

It is well-known that cancellation in short character sums (e.g. Burgess' estimates) yields bounds on the least quadratic nonresidue. Scant progress has been made on short character sums since Burgess' work, so it is desirable to find another approach to nonresidues. In this note we formulate a new line of attack on the least nonresidue via long character sums, an active area of research. Among other results, we demonstrate that improving the constant in the Pólya-Vinogradov inequality would lead to significant progress on nonresidues. Moreover, conditionally on a conjecture on long character sums, we show that the least nonresidue for any odd primitive character (mod k) is bounded by (log k) 1.4 .

Abstract.A classical result of Paley shows that there are infinitely many quadratic characters χ (mod q) whose character sums get as large as √ q log log q; this implies that a conditional upper bound of Montgomery and Vaughan cannot be improved. In this paper, we derive analogous lower bounds on character sums for characters of odd order, which are best possible in view of the corresponding conditional upper bounds recently obtained by the first author.