The frequency and the structure of large character sums

Abstract

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)|.

On the constant in the Pólya-Vinogradov inequality

“…Proof. Let R = (log q) 3 , and apply Dirichlet's theorem to obtain integers r and k satisfying 1 k N/R, (r, k) = 1 and…”

“…Concerning long character sums, Hildebrand [16] has shown that if χ(−1) = 1 then |S(χ, αq)| < εq 1/2 log q, for all α ∈ (0, 1) except for a set of measure q −c 1 ε and that if α = o(1) and χ(−1) = 1 then S(χ, αq) = o(q 1/2 log q). [1] and Bober, Goldmakher, Granville and Koukoulopoulos [3] have obtained much more precise results concerning the distribution of long character sums and Granville and Soundararajan [13] have obtained results concerning the distribution of short character sums. Granville and Soundararajan [14] have also shown that…”

Large Values of for Th Order Characters and Applications to Character Sums

“…Proof of Theorem 1.2. Let z = (log Q) 5 and let 2 y (log Q) 3/2 be a real number to be chosen later. Then, by Proposition 2.4, it follows that for all but at most Q 1/2 primitive characters χ (mod q) with q Q,…”

“…In a recent work, Bober et al [5] studied the distribution of large values of as varies over non‐principal characters modulo , where is a large prime. Among their results, they showed that the bounds in (1.7) are the best possible.…”

“…In order to establish this result, we relate to via the following bounds, which are valid if the order of is even where is the Legendre symbol modulo three. When is even, this bound is proved in [5, §4]. When is odd, the corresponding bound follows from the pointwise estimate (see, for example, [19, Theorem 9.21]) where is the Gauss sum associated to , which satisfies .…”